2015
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Cover vol. 12, no.2, April 2015
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A TS Fuzzy Model Derived from a Typical MultiLayer Perceptron
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In this paper, we introduce a TakagiSugeno (TS) fuzzy model which is derived from a typical MultiLayer Perceptron Neural Network (MLP NN). At first, it is shown that the considered MLP NN can be interpreted as a variety of TS fuzzy model. It is discussed that the utilized Membership Function (MF) in such TS fuzzy model, despite its flexible structure, has some major restrictions. After modifying the MF, we introduce a TS fuzzy model whose MFs are tunable near and far from focal points, separately. To identify such TS fuzzy model, an incremental learning algorithm, based on an efficient space partitioning technique, is proposed. Through an illustrative example, the methodology of the learning algorithm is explained. Next, through two case studies: approximation of a nonlinear function for a sun sensor and identification of a pH neutralization process, the superiority of the introduced TS fuzzy model in comparison to some other TS fuzzy models and MLP NN is shown.
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A.
Kalhor
System Engineering and Mechatronics Group, Faculty of New Sciences
and Technologies, University of Tehran, Tehran, Iran
System Engineering and Mechatronics Group,
Iran
akalhor@ut.ac.ir


B. N.
Aarabi
Control and Intelligent Processing Center of Excellence, School of
Electrical and Computer Engineering, University of Tehran, Tehran, Iran
Control and Intelligent Processing Center
Iran
araabi@ut.ac.ir


C.
Lucas
Control and Intelligent Processing Center of Excellence, School of
Electrical and Computer Engineering, University of Tehran, Tehran, Iran
Control and Intelligent Processing Center
Iran
lucas@ut.ac.ir


B.
Tarvirdizadeh
System Engineering and Mechatronics Group, Faculty of New Sci
ences and Technologies, University of Tehran, Tehran, Iran
System Engineering and Mechatronics Group,
Iran
bahram@ut.ac.ir
TakagiSugeno fuzzy model
Multi layer perceptron
Tunable membership functions
Nonlinear function approximation
pH neutralization process
[[1] P. Angelov and X. Zhou, On line learning fuzzy rulebased system structure from data##streams, In 2008 IEEE International Conference on Fuzzy Systems within the IEEE World##Congress on Computational Intelligence, Hong Kong., (2008), 915922. ##[2] J. F. Baldwin and S. B. Kandarake, Asymmetric triangular fuzzy sets for classication model,##In Lecture Notesin Articial Intelligent, (2003), 364370.##[3] J. N. Choi, S. K. Oh and W. Pedrycz., Identication of fuzzy models using a successive tuning##method with a variant identication ratio, Fuzzy Sets and Systems, 159 (2008), 28732889.##[4] B. L. R. De Moor, ed., DaISy: Database for the Identication of Systems, De##partment of Electrical Engineering, 2008, ESAT/SISTA, K.U.Leuven, Belgium, URL:##http://homes.esat.kuleuven.be/~smc/daisy/. (visited on Oct. 10, 2010).##[5] H. Du and N. Zhang, Application of evolving TakagiSugeno fuzzy model to nonlinear system##identication, Applied soft computing, 8 (2007), 676686.##[6] A. Fiordaliso, A constrained TakagiSugeno fuzzy system that allows for better interpretation##and analysis, Fuzzy Sets and Systems, 118 (2001), 307318.##[7] M. Hell, S. P. Campinas, R. Ballini, Jr. P. Costa and F. Gomid, Training neurofuzzy networks##with participatory learning, In: proceeding of 5th Conference of the EUSFLAT, (2007), 231##[8] J. S. R. Jang, ANFIS: Adaptivenetworkbased fuzzy inference system, IEEE Trans. Syst.##Man and Cybern, 23 (1993), 665685.##[9] A. Kalhor, B. N. Araabi and C. Lucas, A new systematic design for habitually linear evolving##TS fuzzy model, Journal of Expert systems with applications, 39 (2012), 17251736.##[10] A. Kalhor, B. N. Araabi and C. Lucas, An online predictor model as adaptive habitually##linear and transiently nonlinear model, Evolving Systems, 1 (2010), 2941.##[11] A. Kalhor, B. N. Araabi and C. Lucas, A new highorder TakagiSugeno fuzzy model based on##deformed linear models, Amirkabir Int. J. of Modeling Identication, Simulation and Control,##42 (2010), 4352.##[12] A. Kalhor, B. N. Araabi and C. Lucas, Reducing the number of local linear models in neuro{##fuzzy modeling: A splitandmerge clustering approach, Applied Soft Computing Journal, 11##(2011), 5582{5589.##[13] N. Kasabov, DENFIS: Dynamic Evolving NeuralFuzzy Inference System and its application##for timeseries prediction, IEEE Trans. Fuzzy Syst., 10 (2012) 144154.##[14] V. Krkov: Kolmogorov's theorem and multilayer neural networks, Neural networks 5: 501##506, 1992.##[15] D. H. Lee, Y. H. Joo and M. H. Tak, Local stability analysis of continuoustime Takagi{##Sugeno fuzzy systems: A fuzzy Lyapunov function approach, Information Sciences, 257##(2014), 163175.##[16] C. H. Lee and H. Y. Pan, Performance enhancement for neural fuzzy systems using asym##metric membership functions, Fuzzy Sets and Systems, 160 (2009), 949971.##[17] C. H. Lee and C. C. Teng, Fine tuning of membership functions for fuzzy neural systems,##Asian J. Control, 3 (2001), 216225.##[18] G. Leng, T. M. Mc Ginnity and G. Prasad, An approach for online extraction of fuzzy rules##using a selforganising fuzzy neural network, Fuzzy Sets and Systems, 150 (2005), 211243.##[19] C. Li C, K. H. Cheng and J. D. Lee, Hybrid learning neuro fuzzy approach for complex##modeling using asymmetric fuzzy sets, In: Proc. of the17th IEEE International Conf. on##Tools with Articial Intelligence, (2005), 397401.##[20] C. Li, J. Zhou, X. Xiang, Q. Li and X. An, TS fuzzy model identication based on a novel##fuzzy cregression model clustering algorithm, Engineering Applications of Articial Intelligence,##22 (2009), 646653.##[21] C. J. Lin and W. H. Ho, An asymmetricsimilaritymeasurebased neural fuzzy inference##system, Fuzzy Sets and Systems, 152 (2005), 535{551.##[22] T. J. Mc Avoy, E. Hsu and S. Lowenthal, Dynamics of pH in controlled stirred tank reactor,##Ind. Eng. Chem. Process Des. Develop., 11 (1972), 7178.##[23] O. Nelles, Nonlinear System Identication, In: New York: Springer, (2001), Section 13.3.1,##[24] P. Nikdel, M. Hosseinpour, M. A. Badamchizadeh and M. A. Akbari, Improved Takagi{##Sugeno fuzzy modelbased control of ##exible joint robot via HybridTaguchi genetic algorithm,##Engineering Applications of Articial Intelligence, 33 (2014), 1220. ##[25] J. Park and I. W. Sandberg, Universal Approximation Using RadialBasisFunction Net##works, Neural Computation, 3 (1991), 246257.##[26] K. B. Petersen and M. S. Pedersen, The matrix cookbook, http://matrixcookbook.com, Version:##Nov. 14, 2008.##[27] Y. Shi, R. Eberhart and Y. Chen, Implementation of evolutionary fuzzy systems, IEEE Trans.##on Fuzzy Sys., 7 (1999), 109118.##[28] T. Takagi and M. Sugeno, Fuzzy identication of systems and its applications to modeling##and control, IEEE Trans. Syst., Man, and Cybern., 15 (1985), 116132.##[29] D. Wang, C. Quek and G. S. Ng, MSTSKfnn: Novel TakagiSugenoKang fuzzy neural##network using ART like clustering., In: proceeding of IEEE international joint conference on##Neural Networks., (2004), 23612366.##[30] X. Xie, L. Lin and S. Zhong, Process Takagi{Sugeno model: A novel approach for han##dling continuous input and output functions and its application to time series prediction,##KnowledgeBased Systems, 63 (2014), 4658.##[31] H. Ying, General SISO Takagi{Sugeno fuzzy systems with linear rule consequent are univer##sal approximators, IEEE Trans. on Fuzzy Systems, 6 (1998), 582587.##]
Modeling of Epistemic Uncertainty in Reliability Analysis of Structures Using a Robust Genetic Algorithm
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In this paper the fuzzy structural reliability index was determined through modeling epistemic uncertainty arising from ambiguity in statistical parameters of random variables. The First Order Reliability Method (FORM) has been used and a robust genetic algorithm in the alpha level optimization method has been proposed for the determination of the fuzzy reliability index. The sensitivity level of fuzzy response due to the introduced epistemic uncertainty was also measured using the modified criterion of Shannon entropy. By introducing bounds of uncertainty, the fuzzy response obtained from the proposed method presented more realistic estimation of the structure reliability compared to classic methods. This uncertainty interval is of special importance in concrete structures since the quality of production and implementation of concrete varies in different cross sections in reality. The proposed method is implementable in reliability problems in which most of random variables are fuzzy sets and in problems containing nonlinear limit state functions and provides a precise acceptable response. The capabilities of the proposed method were demonstrated using different examples. The results indicated the accuracy of the proposed method and showed that classical methods like FORM cover only special case of the proposed method.
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Mansour
Bagheri
Department of Civil Engineering, University of Sistan and Baluchestan, Zahedan, Iran
Department of Civil Engineering, University
Iran
mnsrbagheri@gmail.com


Mahmoud
Miri
Department of Civil Engineering, University of Sistan and Baluchestan, Zahedan, Iran
Department of Civil Engineering, University
Iran
mmiri@eng.usb.ac.ir


Naser
Shabakhty
Department of Civil Engineering, University of Sistan and Baluchestan, Zahedan, Iran
Department of Civil Engineering, University
Iran
shabakhty@eng.usb.ac.ir
Fuzzy reliability index
Alpha level optimization method
Genetic algorithm
First order reliability method
[[1] G. Alefeld and J. Hertzberger, Introduction to interval computations, Academic Press, New##York, 1983.##[2] D. K. Armen and D. Taleen, Multiple design points in rst and secondorder reliability,##Journal of Structural Safety, 20 (1998), 3749.##[3] M. S. Arumugam, M. V. C. Rao and R. Palaniappan, New hybrid genetic operators for real##coded genetic algorithm to compute optimal control of a class of hybrid systems, Journal of##Applied Soft Computing, 6 (2005), 3852.##[4] S. K. Au and J. L. Beck,Estimation of small failure probabilities in high dimension by subset##simulation, Journal of Probabilistic Engineering Mechanics, 16(4) (2001), 263277.##[5] B. M. Ayyub and R. H. McCuen, Probability, statistics, and reliability for engineers and##scientists, 2nd Ed. Chapman Hall/CRC, Boca Raton, Fla, 2003.##[6] Y. BenHaim, Robust reliability in mechanical sciences, Springer, Berlin, 1996.##[7] K. Breitung, Asymptotic approximations for multinormal integrals, Journal of Engineering##Mechanics ASCE, 110(3) (1984), 357366.##[8] A. DerKiureghian and O. Ditlevsen, Aleatory or epistemic? Does it matter?, Journal of##Structural Safety, 31 (2009), 105112.##[9] W. Dong and H. Shah, Vertex method for computing functions on fuzzy variables, Journal of##Fuzzy Sets and Systems, 24(1) (1978), 6578.##[10] A. H Elhewy, E. Mesbahi and Y. Pu, Reliability analysis of structures using neural network##method, Journal of Probability Engineering Mechanics, 21 (2006), 4453.##[11] S. Freitag, W. Graf and M. Kaliske, A material description based on recurrent neural networks##for fuzzy data and its application within the nite element method, Journal of Computers##and Structures, 124 (2013), 2937.##[12] M. Giuseppe and Q. Giuseppe, A new possibilistic reliability index denition, Journal of##ActaMechanica, 210 (2010), 291303.##[13] D. E. Goldberg, Genetic Algorithms in search, optimization and machine learning, Addison##Wesley, 1989.##[14] F. Grooteman, Adaptive radialbased importance sampling method for structural reliability,##Journal of Structural Safety, 30(6) (2008), 533542.##[15] J. W. Hall and J. Lawry, Fuzzy label methods for constructing imprecise limit state functions,##Journal of Structural Safety, 25(4) (2003), 317341.##[16] M. Hanss and S. Turrin, A fuzzybased approach to comprehensive modeling and analysis of##systems with epistemic uncertainties, Journal of Structural Safety, 32(6) (2010), 433441. ##[17] A. M. Hasofer and N. C. Lind, Exact and invariant second moment code format, Journal of##Engineering Mechanics ASCE, 100 (1974), 111121.##[18] J. C. Helton and W. L. Oberkampf, alternative representations of epistemic uncertainty,##Journal of Reliability Engineering and System Safety, 85(13) (2004), 110.##[19] J. E. Hurtado and D. A. Alvarez,Neural networkbased reliability analysis: a comparative##study, Journal of Computer Methods in Applied Mechanics and Engineering, 191(12)##(2001), 113132.##[20] J. E. Hurtado, D. A. Alvarez and J. Ramirez, Fuzzy structural analysis based on fundamental##reliability concepts, Journal of Computers and Structures, 112 (2012), 183192.##[21] P. Inseok and V. Ramana V Grandhi, Quantication of modelform and parametric uncer##tainty using evidence theory, Journal of Structural Safety, 39 (2012), 4451.##[22] F. Jalayer, I. Iervolino and G. Manfredi, Structural modeling uncertainties and their in##on seismic assessment of existing RC structures, Journal of Structural Safety, 32(3) (2010),##[23] H. Jinsong and D. V. Griths, Observations on FORM in a simple geomechanics example,##Journal of Structural Safety, 33(1) (2011), 115119.##[24] S. D. Koduru and T. Haukaas, Feasibility of FORM in nite element reliability analysis,##Journal of Structural Safety, 32(2) (2010), 145153.##[25] H. Kwakernaak, Fuzzy random variables  I. Denitions and Theorems, Journal of Informa##tion Science, 15 (1978), 129.##[26] H. Kwakernaak, Fuzzy random variables  II.Algorithms and Examples for the Discrete Case,##Journal of Information Science, 17 (1979), 253278.##[27] H. O. Madsen, S. Krenk and N. C. Lind, Methods of structural safety, New York: Dover##Publications, 2006.##[28] F. Massa F, K. Run, T. Tison and B. Lallemand, A complete method for ecient fuzzy##modal analysis, Journal of Sound and Vibration, 309(12) (2008), 6385.##[29] R. E. Melchers, Structural reliability analysis and prediction, 2nd Ed. Chichester John Wiley##and Sons, 1999.##[30] R. E. Melchers, M. Ahammed and C. Middleton, FORM for discontinuous and truncated##probability density functions, Journal of Structural Safety, 25(3) (2003), 305313.##[31] B. Moller and M. Beer, Fuzzy randomness  uncertainty in civil engineering and computa##tional mechanics, Springer Verlag, Berlin, 2004.##[32] B. Moller, W. Graf and M. Beer, Fuzzy structural analysis using level optimization, Journal##of Computational Mechanics, 26(6) (2000), 547565.##[33] B. Moller, W. Graf, M. Beer and R. Schneider,Safety assessment of structures in view of##fuzzy randomness, Journal of Computers and Structeurs, 81 (2003), 15671582.##[34] A. S. Nowak and K. R. Collins, Reliability of structures, McGrawHill, 2000.##[35] M. V. Rama Rao, A. Pownuk, S. Vandewalle and D. Moens, Transient response of structures##with uncertain structural parameters, Journal of Structural Safety, 32 (2010), 449460.##[36] M. Rashki, M. Miri and M. AzhdaryMoghaddam, A new ecient simulation method to##approximate the probability of failure and most probable point, Journal of Structural Safety,##39 (2012), 2229.##[37] T. V. Santosh, R. K. Saraf, A. K. Ghosh and H. S. Kushwaha, Optimum step length selection##rule in modied HLRF method for structural reliability International Journal of Pressure##Vessels and Piping, 83 (2006), 742748.##[38] A. Seranska, M. Kaliske, C. Zopf and W. Graf, A multiobjective optimization approach##with consideration of fuzzy variables applied to structural tire design, Journal of Computers##and Structures, 116(1), (2013), 917.##[39] M. C. Tae and C. L. Byung, Reliabilitybased design optimization using convex linearization##and sequential optimization and reliability assessment method, Journal of Structural Safety,##33(1) (2011), 4250.##[40] J. R. Timothy, (2010), Fuzzy logic with engineering applications, 3rd Ed. Publisher Wiley,##[41] G. Wei, S. Chongmin and TL. Francis, Probabilistic interval analysis for structures with##uncertainty, Journal of Structural Safety, 32(3) (2010), 191199.##[42] M. William and M. Bulleit, Uncertainty in structural engineering, Journal of Practice Peri##odical on Structural Design and Construction, 13(1) (2008), 2430.##[43] L. N. Xing, Y. W. Chen, K. W. Yang, F. Hou, S. Xue and P. C. Huai, A hybrid approach##combining an improved genetic algorithm and optimization strategies for the asymmetric##traveling salesman problem, Journal of Engineering Applications of Articial Intelligence, 21##(2008), 13701380.##[44] L. A. Zadeh, Fuzzy sets, Journal of Information Control, 8(3) (1965), 338553.##[45] H. Zhang, Interval importance sampling method for nite elementbased structural reliability##assessment under parameter uncertainties, Journal of Structural Safety, 38 (2012), 110.##[46] M. Q. Zhang, M. Beer, S. T. Quek and Y. S. Choo, Comparison of uncertainty models##in reliability analysis of oshore structures under marine corrosion, Journal of Structural##Safety, 32 (2010), 425432.##[47] Y. G. Zhao and T. Ono, Moment methods for structural reliability, Journal of Structural##Safety, 23 (2001), 4775.##[48] H. J. Zimmermann, Fuzzy set theory and its applications, Kluwer Academic Publishers, Lon##don, 1992.##]
EQlogics with delta connective
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In this paper we continue development of formal theory of a special class offuzzy logics, called EQlogics. Unlike fuzzy logics being extensions of theMTLlogic in which the basic connective is implication, the basic connective inEQlogics is equivalence. Therefore, a new algebra of truth values calledEQalgebra was developed. This is a lower semilattice with top element endowed with two binaryoperations of fuzzy equality and multiplication. EQalgebra generalizesresiduated lattices, namely, every residuated lattice is an EQalgebra but notviceversa.In this paper, we introduce additional connective $logdelta$ in EQlogics(analogous to Baaz delta connective in MTLalgebra based fuzzy logics) anddemonstrate that the resulting logic has again reasonable properties includingcompleteness. Introducing $Delta$ in EQlogic makes it possible to prove alsogeneralized deduction theorem which otherwise does not hold in EQlogics weakerthan MTLlogic.
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M.
Dyba
University of Ostrava, NSC IT4Innovations, 30. dubna 22, 702 00 Ostrava,
Czech Republic
University of Ostrava, NSC IT4Innovations,
Czech Republic
martin.dyba@osu.cz


V.
Novak
University of Ostrava, NSC IT4Innovations, 30. dubna 22, 702 00 Ostrava,
Czech Republic
University of Ostrava, NSC IT4Innovations,
Czech Republic
vilem.novak@osu.cz
EQalgebra
EQlogic
Equational logic
Delta connective
Generalized deduction theorem
[[1] P. Cintula, P. Hajek, R. Horck, Formal systems of fuzzy logic and their fragments, Annals##of Pure and Applied Logic, 150 (2007), 40{65.##[2] P. Cintula and C. Noguera, A general framework for Mathematical Fuzzy Logic, In: P. Cin##tula, P. Hajek, C. Noguera, eds., Handbook of Mathematical Fuzzy Logic  volume 1, Studies##in Logic, Mathematical Logic and Foundations, vol. 37. College Publications, Londres 2011,##[3] M. Dyba and V. Novak, EQlogics: Noncommutative fuzzy logics based on fuzzy equality,##Fuzzy Sets and Systems, 172 (2011), 13{32.##[4] M. ElZekey, Representable good EQalgebras, Soft Computing, 14 (2009), 1011{1023.##[5] M. ElZekey, V. Novak and R. Mesiar, On good EQalgebras, Fuzzy Sets and Systems, 178##(2011), 1{23.##[6] F. Esteva and L. Godo, Monoidal tnorm based logic: towards a logic for leftcontinuous##tnorms, Fuzzy Sets and Systems, 124 (2001), 271{288.##[7] S. Gottwald, Mathematical fuzzy logics, Bulletin of Symbolic Logic, 14 (2) (2008), 210{239.##[8] S. Gottwald and P. Hajek, Triangular normbased mathematical fuzzy logics, In: E. Kle##ment, R. Mesiar (Eds.), Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular##Norms, Elsevier, Amsterdam, (2005), 257{299.##[9] D. Gries and F. Schneider, A Logical Approach to Discrete Math, SpringerVerlag, Heidelberg,##[10] D. Gries and F. Schneider, Equational propositional logic, Information Processing Letters,##53 (1995), 145{152.##[11] P. Hajek, Metamathematics of Fuzzy Logic, Dordrecht, Kluwer, 1998.##[12] V. Novak, EQalgebras: primary concepts and properties, In: Proc. CzechJapan Seminar,##Ninth Meeting. Kitakyushu& Nagasaki, August 18{22, 2006, Graduate School of Information,##Waseda University, (2006), 219{223.##[13] V. Novak, Which logic is the real fuzzy logic?, Fuzzy Sets and Systems, 157 (2006), 635{641.##[14] V. Novak, EQalgebras in progress, In: O. Castillo, ed., Theoretical Advances and Applica##tions of Fuzzy Logic and Soft Computing, Springer, Berlin, (2007), 876{884.##[15] V. Novak, EQalgebrabased fuzzy type theory and its extensions, Logic Journal of the IGPL,##19 (2011), 512{542.##[16] V. Novak and B. de Baets, EQalgebras, Fuzzy Sets and Systems, 160 (2009), 2956{2978.##[17] G. Tourlakis, Mathematical Logic, New York, J. Wiley & Sons, 2008.##]
Bifuzzy core of fuzzy automata
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2
The purpose of the present work is to introduce the concept of bifuzzy core of a fuzzy automaton, which induces a bifuzzy topology on the stateset of this fuzzy automaton. This is shown that this bifuzzy topology can be used to characterize the concepts such as bifuzzy family of submachines, bifuzzy separable family and bifuzzy retrievable family of a fuzzy automaton.
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S. P.
Tiwari
Department of Applied Mathematics, Indian School of Mines, Dhanbad
826004, India
Department of Applied Mathematics, Indian
India
sptiwarimaths@gmail.com


Anupam K.
Singh
Department of Applied Mathematics, Indian School of Mines,
Dhanbad826004, India
Department of Applied Mathematics, Indian
India
anupam09.bhu@gmail.com


Shambhu
Sharan
Department of Mathematics, School of Applied Sciences, KIIT Uni
versity, Bhubaneswar751024, India
Department of Mathematics, School of Applied
India
ssharanfma@kiit.ac.in


Vijay K.
Yadav
Department of Applied Mathematics, Indian School of Mines, Dhanbad
826004, India
Department of Applied Mathematics, Indian
India
Fuzzy automata
Bifuzzy source
Bifuzzy successor
Bifuzzy core
Bifuzzy topology
[[1] Z. Bavel and J. W. Thomas, On the decomposability of monadic algebras and automata,##Proc. of the 8th Annual Symp. on Switching and Automata Theory, (1967), 322335.##[2] P. Das, A fuzzy topology associated with a fuzzy nite state machine, Fuzzy Sets and Systems,##105(3) (1999), 469479.##[3] W. M. L. Holcombe, Algebraic automata theory, Cambridge University Press, 1982.##[4] J. E. Hopcroft and J. D. Ullman, Introduction to automata theory, languages and computa##tion, AddisonWesley, New York, 1979.##[5] J. Ignjatovic, M. Ciric and V. Simoovic, Fuzzy relation equations and subsystems of fuzzy##transition systems, KnowledgeBased Systems, 38 (2013), 4861.##[6] M. Ito, Algebraic structures of automata, Theoretical Computer Science, 429 (2012), 164168.##[7] J. H. Jin, Q. G. Li and Y. M. Li, Algebraic properties of Lfuzzy nite automata, Information##Sciences, 234 (2013), 182202.##[8] Y. B. Jun, Intuitionistic fuzzy nite state machines, Journal of Applied Mathematics and##Computing, 17 (2005), 109120.##[9] Y. B. Jun, Intuitionistic fuzzy nite switchboard state machines, Journal of Applied Mathe##matics and Computing, 20 (2006), 315325.##[10] Y. B. Jun, Quotient structures of intuitionistic fuzzy nite state machines, Information Sci##ences, 177 (2007), 49774986. ##[11] Y. H. Kim, J. G. Kim and S. J. Cho, Products of Tgeneralized state machines and T##generalized transformation semigroups, Fuzzy Sets and Systems, 93(3) (1998), 8797.##[12] H. V. Kumbhojkar and S. R. Chaudhri, On proper fuzzication of fuzzy nite state machines,##International Journal of Fuzzy Mathematics, 8(4) (2008), 10191027.##[13] W. Lihua and D. Qiu, Automata theory based on complete residuated latticevalued logic:##Reduction and minimization, Fuzzy Sets and Systems, 161(12) (2010), 16351656.##[14] R. Lowen, Fuzzy topological spaces and fuzzy compactness, Journal of Mathematical Analysis##and Applications, 56(3) (1976), 621633.##[15] D. S. Malik, J. N. Mordeson and M. K. Sen, Submachines of fuzzy nite state machine,##Journal of Fuzzy Mathematics, 2(4) (1994), 781792.##[16] J. N. Mordeson and D. S. Malik, Fuzzy automata and languages: theory and applications,##Chapman and Hall/CRC, London/Boca Raton, 2002.##[17] K. Peeva, Finite Lfuzzy acceptors, regular Lfuzzy grammars and syntactic pattern recogni##tion, International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems, 12(1)##(2004), 89104.##[18] D. Qiu, Characterizations of fuzzy nite automata, Fuzzy Sets and Systems, 141(3) (2004),##[19] E. S. Santos, Maximin automata, Information and Control, 12(4) (1968), 367377.##[20] W. Shukla and A. K. Srivastava, A topology for automata: A note, Information and Control,##32(2) (1976), 163168.##[21] A. K. Srivastava and W. Shukla, A topology for automata II, International Journal of Math##ematics and Mathematical Sciences, 9(3) (1986), 425428.##[22] A. K. Srivastava and S. P. Tiwari, A topology for fuzzy automata, Proc. AFSS International##Conference on Fuzzy Systems, Lecture Notes in Articial Intelligence, Springerverlag, 2275##(2002), 485491.##[23] W. G.Wee, On generalizations of adaptive algorithm and application of the fuzzy sets concept##to pattern classication, Ph. D. Thesis, Purdue University, 1967.##[24] M. S. Ying, A new approach for fuzzy topology (I), Fuzzy Sets and Systems, 39(3) (1991),##[25] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338353.##[26] Q. Zhang and Y. Huang, Intuitionistic fuzzy automata based on complete residuated lattice##valued logic, International Journal of Materials and Product Technology, 45 (1/2/3/4)##(2012), 108118.##]
Existence and uniqueness of the solution of nonlinear fuzzy Volterra integral equations
2
2
In this paper the fixed point theorem of Schauder is used to prove the existence of a continuous solution of the nonlinear fuzzy Volterra integral equations. Then using some conditions the uniqueness of the solution is investigated.
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75
86


T.
Allahviranloo
Department of mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
Department of mathematics, Science and Research
Iran
tofigh@allahviranloo.com


P.
Salehi
Department of mathematics, Hamedan Branch, Islamic Azad University,
Hamedan, Iran
Department of mathematics, Hamedan Branch,
Iran
parhamsalehi@rocketmail.com


M.
Nejatiyan
Department of mathematics, Science and Research Branch, Islamic
Azad University, Tehran, Iran
Department of mathematics, Science and Research
Iran
maryamnejatiyan@yahoo.com
Fuzzy numbers
Fuzzy Volterra integral equations
Existence and uniqueness
[[1] G. A. Anastassiou and S. G. Gal, On a fuzzy trigonometric approximation theorem of##Weierstrasstype, J. Fuzzy Math., 9 (3) (2001), 701708.##[2] K. Balachandran and K. Kanagarajan, Existence of solutions of general nonlinear fuzzy##VolterraFredholm integral equations, J. Appl. Math. Stochast. Anal., 3 (2005), 333343.##[3] K. Balachandran and P. Prakash, Existence of solutions of nonlinear fuzzy VolterraFredholm##integral equations, Indian J. Pure Appl. Math., 33 (3) (2002), 329343.##[4] K. Balachandran and P. Prakash, Existence of solutions of nonlinear fuzzy integral equations##in Banach spaces, Libertas Math., 21 (2001), 9197.##[5] J. J. Buckley and T. Feuring, Fuzzy integral equations, J. Fuzzy Math., 10 (2002), 10111024.##[6] D. Dubois and H. Prade, Fundamentals of fuzzy sets, Springer Netherlands Publisher, 2000.##[7] M. Friedman, M. Ma and A. Kandel, On fuzzy integral equations, Fundam. Inform., 37##(1999), 8999. ##[8] S. G. Gal, Approximation theory in fuzzy setting, in: G. A. Anastassiou (Ed.), Handbook##of AnalyticComputational Methods in applied Mathematics, Chapman & Hall, CRC Press,##Boca Raton, London, New York, Washington DC, (2000), (Chapter 13).##[9] D. N. Georgiou and I. E. Kougias, On fuzzy Fredholm and Volterra integral equations, J.##Fuzzy Math., 9 (4) (2001), 943951.##[10] D. N. Georgiou and I. E. Kougias, Bounded solutions for fuzzy integral equation, Int. J. Math.##Math. Sci., 31 (2) (2002), 109114.##[11] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets Syst., 18 (1986), 3143.##[12] A. Karoui and A. Jawahdou, Existence and approximate Lp and continuous solutions of##nonlinear integral equations of the Hammerstein and Volterra types, Appl. Math. Comput.,##216 (2010), 20772091.##[13] J. Mordeson and W. Newman, Fuzzy integral equations, Inform. Sci., 81 (4) (1995), 215229.##[14] J. J. Nieto and R. Rodriguezlopez, Bounded solutions for fuzzy dierential and integral##equations, Chaos Solitons & Fractals, 27 (5) (2006), 13761386.##[15] J. Y. Park and J. U. Jeong, On the existence and uniquness of solutions of fuzzy Volterra##Fredholm integral equation, Fuzzy Sets Syst., 115 (2000), 425431.##[16] J. Y. Park and J. U. Jeong, A note on fuzzy integral equations, Fuzzy Sets Syst., 108 (1999),##[17] J. Y. Park, Y. C. Kwun and J. U. Jeong, Existence of solutions of fuzzy integral equations##in Banach spaces, Fuzzy Sets Syst., 72 (1995), 373378.##[18] J. Y. Park, S. Y. Lee and J. U. Jeong, The approximate solutions of fuzzy functional integral##equation, Fuzzy Sets Syst., 110 (2000), 7990.##[19] C. Wu and Z. Gong, On Henstock integral of fuzzynumbervalueed functions, Fuzzy Sets##Syst., 120 (2001), 523532.##]
Existence and uniqueness of the solution of fuzzyvalued integral equations of mixed type
2
2
In this paper, existence theorems for the fuzzy VolterraFredholm integral equations of mixed type (FVFIEMT) involving fuzzy number valued mappings have been investigated. Then, by using Banach's contraction principle, sufficient conditions for the existence of a unique solution of FVFIEMT are given. Finally, illustrative examples are presented to validate the obtained results.
1

87
94


R.
Ezzati
Department of Mathematics, Karaj Branch, Islamic Azad University,
Karaj, Iran
Department of Mathematics, Karaj Branch,
Iran
ezati@kiau.ac.ir


F.
Mokhtarnejad
Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran
Department of Mathematics, Karaj Branch,
Iran
fa_mokhtar@yahoo.com
Fuzzy VolterraFredholm integral equation
Twodimensional integral equation
Fuzzy integral equations of mixed type
Fuzzy valued function
[[1] S. Abbasbandy, E. Babolian, M. Alavi, Numerical method for solving linear Fredholm integral##equations of the second kind, Chaos Solitons Fractals, 31(1) (2007), 138146.##[2] O. A. Anastassiou and S. G. Gal, On a fuzzy trigonometric approximation theorem of##Weierstrasstype, Journal of Fuzzy Mathmatics, Los Angles, 9(3) (2001), 701708.##[3] M. A. F. Araghi and N. Parandin, Numerical solution of fuzzy Fredholm integral equations##by the Lagrange interpolation based on the extension principle, Soft Computing, 15 (2011),##24492456.##[4] H. Attari and A. Yazdani, A computational method for fuzzy VolterraFredholm integarl equa##tions, Fuzzy Information and Engineering, 2 (2011), 147156.##[5] R. J. Aumann, Integrals of setvalued functions, J. Math. Anal. Appl., 12 (1965), 112.##[6] K. Balachandran and K. Kanagarajan, Existence of solutions of general nonlinear fuzzy Volter##raFredholm integral equations, J. Appl. Math. Stoch. Anal., 3 (2005), 333343.##[7] K. Balachandran and P. Prakash, Existence of solutions of nonlinear fuzzy VolterraFredholm##integral equations, Indian J. Pure Appl. Math., 33 (2002), 329343.##[8] A. M. Bica, Error estimation in the approximation of the solution of nonlinear fuzzy Fredholm##integral equations, Information Sciences, 178 (2008), 12791292.##[9] A. M. Bica and C. Popescu, Approximating the solution of nonlinear Hammerstein fuzzy##integral equations, Fuzzy Sets and Systems, doi.org/10.1016/j.fss.2013.08.005.##[10] D. Dubois and H. Prade, Towards fuzzy dierential calculus, Fuzzy Sets and Systems, 8##(1982), 117.##[11] R. Ezzati and S. Ziari, Numerical solution and error estimation of fuzzy Fredholm integral##equation using fuzzy Bernstein polynomials, Australian Journal of Basic and Applied Sciences,##5(9) (2011), 20722082.##[12] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986),##[13] O. Kaleva, Fuzzy dirential equations, Fuzzy Sets and Systems, 24 (1987), 301317.##[14] A. Molabahrami, A. Shidfar and A. Ghyasi, An analytical method for solving linear Fredholm##fuzzy integral equations of the second kind, Computers and Mathematics with Applications,##61(9) (2011), 27542761. ##[15] J. Mordeson and W. Newman, Fuzzy integral equations, Information Sciences, 87 (1995),##[16] S. Nanda, On integration of fuzzy mapping, Fuzzy Sets and Systems, 32 (1989), 95101.##[17] J. Y. Park, S. J. Lee and J. U. Jeong, On the existence and uniqueness of solutions of fuzzy##VolterraFredholm integral equations, Fuzzy sets and systems, 115 (2000), 425431.##[18] M. L. Puri and D. A. Ralescu, Fuzzy random variables, J. Math. Anal.Appl., 114 (1986),##[19] D. Ralescu and G. Adams, The fuzzy integrals, J. Math. Anal. Appl., 75 (1980), 562570.##[20] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (1987), 319330.##[21] P. V. Subrahmaniam and S. K. Sudarsanam, On some fuzzy functional equations, Fuzzy Sets##and Systems, 64 (1994), 333338.##[22] P. V. Subrahmaniam and S. K. Sudarsanam, A note on fuzzy Volterra integral equations,##Fuzzy Sets and Systems, 81 (1996), 237240.##[23] M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. Dissertation, Tokyo Inst. of##Tech., 1974.##[24] C. Wu and Z. Gong, On Henstock integral of fuzzynumber valued functions, Fuzzy sets and##systems, 120 (2001), 523532.##[25] Z. Wang, The autocontinuity of setfunction and the fuzzy integral, J. Math. Anal. Appl., 99##(1984), 195218.##[26] H. C.Wu, The fuzzy Riemann integral and its numerical integration, Fuzzy Sets and Systems,##110 (2000), 125.##[27] L. A. Zadeh, A fuzzysettheoretic interpretation of linguistic hedges, Journal of Cybernetics,##2 (1972), 434.##[28] L. A. Zadeh, The concept of the linguistic variable and its application to approximate rea##soning, Information Sciences, 8 (1975), 199249.##[29] S. Ziari, R. Ezzati and S. Abbasbandy, Numerical solution of linear fuzzy fredholm inte##gral equations of the second kind using fuzzy haar wavelet, Communications in Computer and##Information Science, 299 (2012), 7989.##]
Fuzzy resolvent equation with $H(cdot,cdot)$$phi$$eta$accretive operator in Banach spaces
2
2
In this paper, we introduce and study fuzzy variationallike inclusion, fuzzy resolvent equation and $H(cdot,cdot)$$phi$$eta$accretive operator in real uniformly smooth Banach spaces. It is established that fuzzy variationallike inclusion is equivalent to a fixed point problem as well as to a fuzzy resolvent equation. This equivalence is used to define an iterative algorithm for solving fuzzy resolvent equation. Some examples are given.
1

95
106


Rais
Ahmad
Department of Mathematics, Aligarh Muslim University, Aligarh
202002, India
Department of Mathematics, Aligarh Muslim
India
raisain_123@rediffmail.com


Mohd
Dilshad
Department of Mathematics, Aligarh Muslim University, Aligarh
202002, India
Department of Mathematics, Aligarh Muslim
India
mdilshaad@gmail.com
Fuzzy variationallike inclusion
Fuzzy resolvent equation
$H(cdot
cdot)$$phi$$eta$accretive operator
Algorithm
Fixed point
[[1] Q. H. Ansari, Certain problems concerning variational inequalities, Ph.D Thesis, Aligarh##Muslim University, Aligarh, India, 1988.##[2] S. S. Chang, Fuzzy quasivariational inclusions in Banach spaces, Appl. Math. Comput., 145##(2003), 805819.##[3] S. S. Chang, Y. Zhu, On variational inequalities for fuzzy mappings, Fuzzy Sets and Systems,##32 (1989), 359367.##[4] H. X. Dai, Generalized mixed variationallike inequalities with fuzzy mappings, J. Comput.##Appl. Math., 224 (2009), 2028.##[5] X. P. Ding and J. Y. Park, A new class of generalized nonlinear implicit quasivariational##inclusions with fuzzy mappings, J. Comp. Math. Appl., 138 (2002) 249257.##[6] X. P. Ding, Algorithm of solutions for mixed implicit quasivariational inequalities with fuzzy##mappings, Comput. Math. Appl., 38 (1999), 231241. ##[7] C. F. Hu, Solving variational inequalities in fuzzy environment, J. Math. Anal., 249 (2000),##[8] P. Kumam and N. Petrol, Mixed variationallike inequality for fuzzy mappings in re##Banach spaces, J. Inequal. Appl., 2009 (2009), 115.##[9] Z. Liu, L. Debnath, S. M. Kang and J. S. Ume, Generalized mixed quasivariational inclusions##and generalized mixed resolvent equations for fuzzy mappings, Appl. Math. Comput., 149##(2004), 879891.##[10] B. S. Lee, M. F. Khan, Salahuddin, Fuzzy nonlinear setvalued variational inclusions, Com##put. Math. Appl., 60 (2010), 17681775.##[11] Jr. S. B. Nadler, Multivalued contraction mappings, Pacic. J. Math., 30 (1969), 475488.##[12] M. A. Noor, Variational inequalities for fuzzy mappings (II), Fuzzy Sets and System, 110##(2000) 101108.##[13] Z. Wu and J. Xu, Generalized convex fuzzy mappings and fuzzy variationallike inequalities,##Fuzzy Sets and Systems, 160(11) (2009), 15901619.##[14] H. K. Xu, Inequalities in Banach spaces and applications, Nonlinear Analysis, Theory Meth##ods and Applications, 16(12) (1991), 11271138.##[15] L. A. Zadeh, Fuzzy Sets, Inform. Contr., 8 (1965), 338353.##[16] Y. Z. Zou and N. J. Huang, H(; )accretive operator with an application for solving varia##tional inclusions in Banach spaces, Appl. Math. Comput., 204 (2008), 809816.##]
Classifying fuzzy normal subgroups of finite groups
2
2
In this paper a first step in classifying the fuzzy normalsubgroups of a finite group is made. Explicit formulas for thenumber of distinct fuzzy normal subgroups are obtained in theparticular cases of symmetric groups and dihedral groups.
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107
115


Marius
Tarnauceanu
Faculty of Mathematics, "Al.I. Cuza" University, Iasi, Romania
Faculty of Mathematics, "Al.I. Cuza"
Romania
tarnauc@uaic.ro
Fuzzy normal subgroups
Chains of normal subgroups
Maximal chains of normal subgroups
Symmetric groups
Dihedral groups
[[1] N. Ajmal and K.V. Thomas, The lattice of fuzzy normal subgroups is modular, Inform. Sci.##83 (1995), 199{209.##[2] G. Gratzer, General lattice theory, Academic Press, New York, 1978.##[3] T. Head, A metatheorem for deriving fuzzy theorems from crisp versions, Fuzzy Sets and##Systems 73 (1995), 349358; 79 (1996), 277{278.##[4] M. Mashinchi and M. Mukaidono, A classication of fuzzy subgroups, Ninth Fuzzy System##Symposium, Sapporo, Japan, (1992), 649{652.##[5] M. Mashinchi and M. Mukaidono, On fuzzy subgroups classication, Research Report of Meiji##Univ., 9 (1993), 31{36.##[6] J. N. Mordeson, K. R. Bhutani and A. Rosenfeld, Fuzzy group theory, Springer Verlag, Berlin,##[7] V. Murali and B. B. Makamba, Normality and congruence in fuzzy subgroups, Inform. Sci.,##59 (1992), 121{129.##[8] V. Murali and B. B. Makamba, On an equivalence of fuzzy subgroups, I, Fuzzy Sets and##Systems, 123 (2001), 259{264.##[9] V. Murali and B. B. Makamba, On an equivalence of fuzzy subgroups, II, Fuzzy Sets and##Systems, 136 (2003), 93{104.##[10] V. Murali and B. B. Makamba, On an equivalence of fuzzy subgroups, III, Int. J. Math. Sci.,##36 (2003), 2303{2313.##[11] V. Murali and B. B. Makamba, Counting the number of fuzzy subgroups of an abelian group##of order pnqm, Fuzzy Sets and Systems, 144 (2004), 459{470.##[12] V. Murali and B. B. Makamba, Fuzzy subgroups of nite abelian groups, FJMS, 14 (2004),##[13] R. Schmidt, Subgroup lattices of groups, de Gruyter Expositions in Mathematics 14, de##Gruyter, Berlin, 1994.##[14] M. Suzuki, Group theory, I, II, Springer Verlag, Berlin, (1982), (1986).##[15] M. Stefanescu and M. Tarnauceanu, Counting maximal chains of subgroups of nite nilpotent##groups, Carpathian J. Math., 25 (2009), 119{127.##[16] M. Tarnauceanu and L. Bentea, On the number of fuzzy subgroups of nite abelian groups,##Fuzzy Sets and Systems, doi: 10.1016/j.fss.2007.11.014, 159 (2008), 1084{1096.##[17] A. C. Volf, Counting fuzzy subgroups and chains of subgroups, Fuzzy Systems & Articial##Intelligence, 10 (2004), 191{200.##]
Numerical solutions of nonlinear fuzzy Fredholm integrodifferential equations of the second kind
2
2
In this paper, we use parametric form of fuzzy number, then aniterative approach for obtaining approximate solution for a classof nonlinear fuzzy Fredholmintegrodifferential equation of the second kindis proposed. This paper presents a method based on NewtonCotesmethods with positive coefficient. Then we obtain approximatesolution of the nonlinear fuzzy integrodifferential equations by an iterativeapproach.
1

117
127


M.
Mosleh
Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran
Department of Mathematics, Firoozkooh Branch,
Iran
mosleh@iaufb.ac.ir


M.
Otadi
Department of Mathematics, Firoozkooh Branch, Islamic Azad University,
Firoozkooh, Iran
Department of Mathematics, Firoozkooh Branch,
Iran
mahmoodotadi@yahoo.com
Nonlinear fuzzy integrodifferential equations
NewtonCotes methods
[[1] S. Abbasbandy and T. Allahviranloo,Numerical solution of fuzzy dierential equation by##RungeKutta method, Nonlinear studies, 11(1) (2004), 117129.##[2] S. Abbasbandy, T. Allaviranloo, O. LopezPouso and J. J. Nieto, Numerical methods for##fuzzy dierential inclusions, Computers & mathematics with applications, 48(1011) (2004),##16331641.##[3] S. Abbasbandy and B. Asady, Newtons method for solving fuzzy nonlinear equations, Applied##Mathematics and Computation, 159(2) (2004), 349356.##[4] S. Abbasbandy, E. Babolian and M. Alavi, Numerical method for solving linear Fredholm##fuzzy integral equations of the second kind, Chaos Solitons & Fractals, 31(1) (2007), 138##[5] S. Abbasbandy and A. Jafarian, Steepest descent method for solving fuzzy nonlinear equa##tions, Applied Mathematics and Computation, 175(1) (2006), 581589.##[6] S. Abbasbandy, J. J. Nieto and M. Alavi, Tuning of reachable set in one dimensional fuzzy##dierential inclusions, Chaos, Solitons & Fractals, 26(5) (2005), 13371341. ##[7] T. Allahviranloo, S. Abbasbandy, N. Ahmady and E. Ahmady, Improved predictorcorrector##method for solving fuzzy initial value problems, Information Sciences, 179(7) (2009), 945955.##[8] T. Allahviranloo, N. Ahmady and E. Ahmady, Numerical solution of fuzzy dierential equa##tions by predictorcorrector method, Information Sciences, 177(7) (2007), 16331647.##[9] T. Allahviranloo, N. A. Kiani and M. Barkhordari,Toward the existence and uniqueness of##solutions of secondorder fuzzy dierential equations, Information Sciences, 179(8) (2009),##12071215.##[10] T. Allahviranloo, N. A. Kiani and N. Motamedi, Solving fuzzy dierential equations by dif##ferential transformation method, Information Sciences, 179(7) (2009), 956966.##[11] K. E. Atkinson,An introduction to numerical analysis, New York: Wiley, 1987.##[12] E. Babolian, H. S. Goghary and S. Abbasbandy,Numerical solution of linear Fredholm fuzzy##integral equations of the second kind by Adomian method, Applied Mathematics and Com##putation, 161(3) (2005), 733744.##[13] C. T. H. Baker, A perspective on the numerical treatment of Volterra equations, J. Comput.##Appl. Math., 125(12) (2000), 217249.##[14] P. Balasubramaniam and S. Muralisankar, Existence and uniqueness of fuzzy solution for the##nonlinear fuzzy integrodierential equations, Applied mathematics letters, 14(4) (2001),##[15] M. I. Berenguer, D. Gamez, A. I. GarraldaGuillem, M. Ruiz Galan and M. C. Serrano Perez,##Biorthogonal systems for solving Volterra integral equation systems of the second kind, J.##Comput. Appl. Math., 235(7) (2011), 18751883.##[16] A. H. Borzabadi and O. S. Fard, A numerical scheme for a class of nonlinear Fredholm##integral equations of the second kind, Journal of Computational and Applied Mathematics,##232(2) (2009), 449454.##[17] S. S. L. Chang and L. Zadeh,On fuzzy mapping and control, IEEE Trans. System Man##Cybernet, 2(1) (1972), 3034.##[18] Y. Chen and T. Tang, Spectral methods for weakly singular Volterra integral equations with##smooth solutions, J. Comput. Appl. Math., 233(4) (2009), 938950.##[19] D. Dubois and H. Prade, Operations on fuzzy numbers, J. Systems Sci., 9(6) (1978), 613626.##[20] D. Dubois and H. Prade, Towards fuzzy dierential calculus, Fuzzy Sets and Systems, 8(13)##(1982), 17.##[21] M. Friedman, M. Ma and A. Kandel, Numerical solutions of fuzzy dierential and integral##equations, Fuzzy Sets and Systems, 106(1) (1999), 3548.##[22] R. Goetschel and W. Vaxman, Elementary calculus, Fuzzy sets and Systems, 18(1) (1986),##[23] H. Hochstadt, Integral equations, New York: Wiley, 1973.##[24] A. Kaufmann and M. M. Gupta, Introduction Fuzzy Arithmetic, Van Nostrand Reinhold,##New York, 1985.##[25] O. Kaleva, Fuzzy dierential equations, Fuzzy Sets and Systems, 24(3) (1987), 301317.##[26] J. P. Kauthen, Continuous time collocation method for VolterraFredholm integral equations,##Numer. Math., 56(1) (1989), 409424.##[27] G. J. Klir, U. S. Clair and B. Yuan, Fuzzy set theory: foundations and applications, Prentice##Hall Inc., 1997.##[28] H. Kwakernaak, Fuzzy random variables. Part I: denitions and theorems, Information Sci##ences, 15(1) (1978), 129.##[29] P. Linz, Analytical and numerical methods for Volterra equations, SIAM, Philadelphia, PA,##[30] M. T. Malinowski,On random fuzzy dierential equations, Fuzzy Sets and Systems, 160(21)##(2009), 31523165.##[31] M. T. Malinowski, Existence theorems for solutions to random fuzzy dierential equations,##Nonlinear Analysis: Theory, Methods & Applications, 73(6) (2010), 15151532.##[32] M. T. Malinowski, Random fuzzy dierential equations under generalized Lipschitz condition,##Nonlinear Analysis: Real World Applications, 13(2) (2012), 860881. ##[33] M. Mosleh and M. Otadi, Simulation and evaluation of fuzzy dierential equations by fuzzy##neural network, Applied Soft Computing, 12(9) (2012), 28172827.##[34] M. Mosleh and M. Otadi, Minimal solution of fuzzy linear system of dierential equations,##Neural Computing and Applications, 21(1) (2012), 329336.##[35] M. L. Puri and D. Ralescu, Fuzzy random variables, Journal of Mathematical Analysis and##Applications, 114(2) (1986), 409422.##[36] M. L. Puri and D. Ralescu, Dierentials of fuzzy functions, Journal of Mathematical Analysis##and Applications, 91(2) (1983), 552558.##[37] H. H. Sorkun and S. Yalcinbas, Approximate solutions of linear Volterra integral equation##systems with variable coecients, Applied Mathematical Modelling, 34(11) (2010), 3451##[38] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, SpringerVerlag, New York,##[39] W. Congxin and M. Ming, Embedding problem of fuzzy number space, Fuzzy Sets and Systems,##45(2) (1992), 189202.##[40] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning,##Information Sciences, 8(3) (1975), 199249.##]
Generated $textbf{textit{L}}$subgroup of an $textbf{textit{L}}$group
2
2
In this paper, we extend the construction of a fuzzy subgroup generated by a fuzzy subset to $L$setting. This construction is illustrated by an example. We also prove that for an $L$subset of a group, the subgroup generated by its level subset is the level subset of the subgroup generated by that $L$subset provided the given $L$subset possesses supproperty.
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129
136


Naseem
Ajmal
Department of Mathematics, Zakir Husain Delhi College,, J.N.Marg,
University of Delhi, Delhi110006, India
Department of Mathematics, Zakir Husain Delhi
India
nasajmal@yahoo.com


Iffat
Jahan
Department of Mathematics, Ramjas College,, University of Delhi,,
Delhi110007, India
Department of Mathematics, Ramjas College,,
India
ij.umar@yahoo.com
$L$algebra
$L$subgroup
Normal $L$subgroup
Generated $L$subgroup
[[1] N. Ajmal, Set product and fuzzy subgroups, Proc. IFSA World Cong., Belgium, ( 1991 ), 37.##[2] N. Ajmal, The lattice of fuzzy normal subgroups is modular, Inform. Sci., 83 (34) (1995),##[3] N. Ajmal, Fuzzy groups with sup property, Inform. Sci., 93 (34) (1996), 247264. ##[4] N. Ajmal and K. V. Thomas, Fuzzy lattices, Inform. Sci., 79 (34) (1994), 271291.##[5] N. Ajmal and K. V. Thomas, The lattices of fuzzy ideals of a ring, Fuzzy Sets and Systems,##74 (3) (1995), 371379.##[6] N. Ajmal and K. V. Thomas, A complete study of the lattices of fuzzy congruences and fuzzy##normal subgroups, Inform. Sci., 82 (34) (1995), 197218.##[7] N. Ajmal and K. V. Thomas, The join of fuzzy algebraic substructures of a group and their##lattices, Fuzzy Sets and Systems, 99 (2) (1998), 213224.##[8] N. Ajmal and K. V. Thomas, A new blueprint for fuzzication : An application to lattices##of fuzzy congruences, J. Fuzzy Math., 7 (2) (1999), 499512.##[9] N. Ajmal and K. V. Thomas, Fuzzy latticesI, J. Fuzzy Math., 10 (2) (2002), 255274.##[10] N. Ajmal and K. V. Thomas, Fuzzy lattices II, J. Fuzzy Math., 10 (2) (2002), 275296.##[11] N. Ajmal and Sunil Kumar, Lattices of subalgebras in the category of fuzzy groups, Fuzzy##Mathematics, 10 (2) (2002), 359369.##[12] N. Ajmal and I. Jahan, A study of normal fuzzy subgroups and characteristic fuzzy subgroups##of a fuzzy group, Fuzzy Information and Engineering, 4 (2) (2012), 123143.##[13] N. Ajmal and A. Jain, Some constructions of the join of fuzzy subgroups and certain lattices##of fuzzy subgroups with sup property, Inform. Sci., 179 (23) (2009), 40704082.##[14] L. Biacino and G. Gerla, Closure systems and Lsubalgebras, Inform. Sci., 33 (3) (1984),##[15] V. N. Dixit, R. Kumar and N. Ajmal, Level subgroups and union of fuzzy subgroups, Fuzzy##Sets and Systems, 37 (3) (1990), 359371.##[16] J. A. Goguen, L fuzzy sets, J. Math. Anal. Appl., 18 (1) (1967), 145174.##[17] T. Head,A metatheorem for deriving fuzzy theorems from crisp versions, Fuzzy Sets and##Systems, 73 (3) (1995), 349358.##[18] R. Kumar, Fuzzy subgroups, fuzzy ideals and fuzzy cosets : Some properties, Fuzzy Sets and##Systems, 48 (2) (1992), 267271.##[19] W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems, 8 (2) (1982),##[20] M. Mashinchi, S. Salili and M. M. Zahedi, Lattice structures on fuzzy subgroups, Bull. Iranian##Math. Soc., 18 (2) (1992), 1729.##[21] S. Ray, Generated and cyclic fuzzy subgroups, Information Sciences, 69 (3) (1993), 185200.##[22] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35(3)(1971), 512517.##[23] N. Sultana and N. Ajmal, Generated fuzzy subgroups: a modication, Fuzzy Sets Systems,##107 (2) (1999), 241243.##[24] Y. Yu, J. N. Mordeson and S. C. Cheng, Elements of algebra, lecture notes in fuzzy math##ematics and computer science, Center for Research in Fuzzy Mathematics and Computer##Science, Creighton University, USA, 1994.##]
A New Approach to Caristi's Fixed Point Theorem on NonArchimedean Fuzzy Metric Spaces
2
2
In the present paper, we give a new approach to Caristi's fixed pointtheorem on nonArchimedean fuzzy metric spaces. For this we define anordinary metric $d$ using the nonArchimedean fuzzy metric $M$ on a nonemptyset $X$ and we establish some relationship between $(X,d)$ and $(X,M,ast )$%. Hence, we prove our result by considering the original Caristi's fixedpoint theorem.
1

137
143


S.
Sedghi
Department of Mathematics, Qaemshahr Branch, Islamic Azad University,
Qaemshahr, Iran
Department of Mathematics, Qaemshahr Branch,
Iran
sedghi.gh@qaemshahriau.ac.ir


N.
Shobkolaei
Department of Mathematics, Babol Branch, Islamic Azad University,
Babol, Iran
Department of Mathematics, Babol Branch,
Iran
nabi_shobe@yahoo.comg


I.
Altun
Department of Mathematics, Faculty of Science and Arts, Kirikkale Uni
versity, 71450 Yahsihan, Kirikkale, Turkey
Department of Mathematics, Faculty of Science
Turkey
ishakaltun@yahoo.com
Fixed point
Caristi map
Fuzzy metric space
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