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Cover vol. 12, no.3, June 2015
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Admissibility analysis for discretetime singular systems with timevarying delays by adopting the statespace TakagiSugeno fuzzy model
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This paper is pertained with the problem of admissibility analysis of uncertain discretetime nonlinear singular systems by adopting the statespace TakagiSugeno fuzzy model with timedelays and normbounded parameter uncertainties. Lyapunov Krasovskii functionals are constructed to obtain delaydependent stability condition in terms of linear matrix inequalities, which is dependent on the lower and upper delay bounds. Finally, numerical examples are provided to substantiate the theoretical results.
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P.
Balasubramaniam pour
Department of Mathematics, Gandhigram Rural Institute  Deemed University, Gandhigram  624 302, Tamilnadu, India
Department of Mathematics, Gandhigram Rural
India
balugru@gmail.com


L.
Jarina Banu
Department of Mathematics, Gandhigram Rural Institute  Deemed
University, Gandhigram  624 302, Tamilnadu, India
Department of Mathematics, Gandhigram Rural
India
ljarina88@gmail.com
Discretetime Singular system
TakagiSugeno fuzzy systems
Stability
LyapunovKrasovskii functional
Linear Matrix Inequality (LMI)
[[1] J. An and G. Wen, Improved stability criteria for timevarying delayed TS fuzzy systems##via delay partitioning approach, Fuzzy Sets Syst., 185(1) (2011), 8394.##[2] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear matrix inequalities in system##and control theory, Soc. Ind. Appl. Math., Philadelphia, 1994. ##[3] I. M. Buzurovic and D. LJ. Debeljkovic, Contact problem and controllability for singular##systems in biomedical robotics, Int. J. Inf. Syst. Sci., 6(2) (2010), 128141.##[4] S. H. Chen and J. H. Chou, Stability robustness of linear discrete singular timedelay systems##with structured parameter uncertainties, IEE Proc. Control Theory Appl., 150(3) (2003),##[5] L. Dai, Singular Control Systems, SpringerVerlag : Berlin, 1989.##[6] Y. Ding, S. Zhong and W. Chen, A delayrangedependent uniformly asymptotic stability##criterion for a class of nonlinear singular systems, Nonlinear Anal. B: Real World Appl.,##12(2) (2011), 11521162.##[7] M. Fang, Delaydependent stabililty analysis for discrete singular systems with timevarying##delays, Acta Automat. Sinica, 36(5) (2010), 751755.##[8] Z. Feng and J. Lam, Robust reliable dissipative ltering for discrete delay singular systems,##Signal Process., 92(12) (2012), 30103025.##[9] C. Huang, Stability analysis of discrete singular fuzzy systems, Fuzzy Sets Syst., 151(1)##(2005), 155165.##[10] J. Jiao, Robust stability and stabilization of discrete singular systems with interval time##varying delay and linear fractional uncertainty, Int. J. Autom. Comput., 9(1) (2012), 815.##[11] F. L. Lewis, A survey of linear singular systems, Circuits Syst. Signal Process., 5(1) (1986),##[12] J. Li, H. Su, Z. Wu and J. Chu, Robust stabilization for discretetime nonlinear Singular##systems with mixed time delays, Asian J. Control, 14(1) (2012), 14111421.##[13] J. Li, H. Su, Z. Wu and J. Chu, Less conservative robust stability criteria for uncertain##discrete stochastic singular systems with timevarying delay, Int. J. Syst. Sci., 44(3) (2013),##[14] J. Lin, S. Fei and J. Shen, Delaydependent H1 ltering for discretetime singular Markovian##jump systems with timevarying delay and partially unknown transition probabilities, Signal##Process., 91(2) (2011), 277289.##[15] I. R. Petersen, A stabilization algorithm for a class of uncertain linear systems, Systems##Control Lett., 8(4) (1987), 351357.##[16] H. Rotstein, M. Sznaier and M. Idan, H2=H1 ltering theory and an aerospace application,##Int. J. Robust Nonlinear Control, 6 (1996), 347366.##[17] T. Takagi and M. Sugeno, Fuzzy identication of systems and its applications to modelling##and control, IEEE Trans. Syst. Man Cybern., 15(1) (1985), 116132.##[18] K. Tanaka and M. Sugeno, Stability analysis and design of fuzzy control systems, Fuzzy Sets##Syst., 45(2) (1992), 135156.##[19] T. Taniguchi, K. Tanaka and H. O. Wang, Fuzzy descriptor systems and nonlinear model##following control, IEEE Trans. Fuzzy Syst., 8(4) (2000), 265452.##[20] Y. Wang, Z. Sun and F. Sun, Robust fuzzy control of a class of nonlinear descriptor systems##with timevarying delay, Int. J. Control Autom. Syst., 2(1) (2004), 7682.##[21] Z.Wu, J. H. Park, H. Su and J. Chu, Admissibility and dissipativity analysis for discretetime##singular systems with mixed timevarying delays, Appl. Math. Comput., 218(13) (2012),##71287138.##[22] Z. Wu, P. Shi, H. Su and J. Chu, Reliable H1 control for discretetime fuzzy systems with##innitedistributed delay, IEEE Trans. Fuzzy Syst., 20(1) (2012), 2231.##[23] Z. Wu, H. Su and J. Chu, Robust exponential stability of uncertain singular markovian jump##timedelay systems, Acta Automat. Sinica, 36(4) (2010), 558563.##[24] Y. Xia, L. Li, M. S. Mahmoud and H. Yang, H1 ltering for nonlinear singular markovian##jumping systems with interval timevarying delays, Int. J. Syst. Sci., 43(2) (2012), 272284.##[25] H. Xin, D. Gan, M. Huang and K. Wang, Estimating the stability region of singular perturba##tion power systems with saturation nonlinearities: an linear matrix inequalitybased method,##IET Control Theory Appl., 4(3) (2010), 351361.##[26] S. Xu, P. V. Dooren, R. Stefan and J. Lam, Robust stability and stabilization for singular##systems with state delay and parameter uncertainty, IEEE Trans. Autom. Control, 47(7)##(2002), 11221128. ##[27] S. Xu, B. Song, J. Lu and J. Lam, Robust stability of uncertain discretetime singular fuzzy##systems, Fuzzy Sets Syst., 158(20) (2007), 23062316.##[28] Q. L. Zhang, Decentralized control and robust control for singular systems, Xian : North##western University Press, 1997.##[29] J. Zhang and Y. Zhao, Asymptotic stability of nonlinear singular discrete systems, Proc.##IEEE Int. Conf. Multimedia Tech., (2011), 24112413.##[30] S. Zhao, Quadratic stabilization for a class of switched Nonlinear singular systems, Int. J.##Inf. Syst. Sci., 5(34) (2009), 425429.##]
Fuzzy Risk Analysis Based on Ranking of Fuzzy Numbers Via New Magnitude Method
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Ranking fuzzy numbers plays a main role in many applied models inreal world and in particular decisionmaking procedures. In manyproposed methods by other researchers may exist some shortcoming.The most commonly used approaches for ranking fuzzy numbers isbased on defuzzification method. Many ranking fuzzy numberscannot discriminate between two symmetric fuzzy numbers withidentical core. In 2009, Abbasbandy and Hajjari proposed anapproach for ranking normal trapezoidal fuzzy numbers, whichcomputed the magnitude of fuzzy numbers namely ``Mag" method.Then Hajjari extended it for nonnormal trapezoidal fuzzy numbersand also for all generalized fuzzy numbers. However, thesemethods have the weakness that we mentioned above. Moreover, theresult is not consistent with human intuition in this case.Therefore, we are going to present a new method to overcome thementioned weakness. In order to overcome the shortcoming, a newmagnitude approach for ranking trapezoidal fuzzy numbers based onminimum and maximum points and the value of fuzzy numbers isgiven. The new method is illustrated by some numerical examplesand in particular, the results of ranking by the proposed methodand some common and existing methods for ranking fuzzy numbers iscompared to verify the advantages of presented method.
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T.
Hajjari
Mathematics Department, Firoozkooh Branch of Islamic Azad University,
Firoozkooh, Iran
Mathematics Department, Firoozkooh Branch
Iran
tayebehajjari@iaufb.ac.ir
DecisionMaking
Magnitude
Fuzzy numbers
Ranking
[[1] S. Abbasbandy and B. Asady, Ranking of fuzzy numbers by sign distance, Inform. Sci., 176##(2006), 24052416.##[2] S. Abbasbandy and T. Hajjari, A new approach for ranking of trapezoidal fuzzy numbers,##Comput. Math. Appl., 57 (2009), 413419. ##[3] S. Abbasbandy and T. Hajjari, An improvement on centroid point method for ranking of##fuzzy numbers, J. Sci. I.A.U., 78 (2011), 109119.##[4] M. Adamo, Fuzzy decision trees, Fuzzy Sets and Systems, 4 (1980), 207219.##[5] B. Asady, The revised method of ranking LR fuzzy number based on deviation degree, Expert##Syst with Applications, 37 (2010), 50565060.##[6] J. F. Baldwin and N. C. F. Guild, Comparison of fuzzy numbers on the same decision space,##Fuzzy Sets and Systems, 2 (1979), 213233.##[7] S. Bass and H. Kwakernaak,Rating and ranking of multipleaspect alternatives using fuzzy##sets, Automatica, 13 (1977), 4758.##[8] G. Bortolan and R. Degani, A review of some methods for ranking fuzzy numbers, Fuzzy Sets##and Systems, 15 (1985), 119.##[9] S. H. Chen, Ranking fuzzy numbers with maximizing set and minimizing set, Fuzzy Sets and##Systems, 17 (1985), 113129.##[10] W. K. Chang, Ranking of fuzzy utilities with triangular membership functions, International##Conference on Plicy Analysis and Informations Systems, Tamkang University, R. O. C.,##(1981), 163171.##[11] C. H. Cheng, A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and##Systems, 95 (1998), 307317.##[12] S. J. Chen and S. M. Chen, Fuzzy risk analysis based on ranking of generalized trapezoidal##fuzzy numbers, Applied Intelligence, 26 (2007), 111.##[13] S. M. Chen and J. H. Chen, Fuzzy risk analysis based on ranking generalized fuzzy numbers##with dierent heights and dierent spreads, Expert Systs with Applications, 36 (2009), 6833##[14] S. J Chen and C. L. Hwang, Fuzzy multiple attribute decision making, SpingerVerlag, Berlin,##[15] S. M. Chen and K. Sanguansat, Analysing fuzzy risk based on a new fuzzy ranking generalized##fuzzy numbers with dierent heights and dierent spreads, Expert Systs with Applications,##38 (2011), 21632171.##[16] C. C. Chen and H. C. Tang, Ranking nonnormal pnorm trapezoidal fuzzy numbers with##integral value, Comput. Math. Appl., 56 (2008), 23402346.##[17] F. Choobineh and H. Li, An index for ordering fuzzy numbers, Fuzzy Sets and Systems, 54##(1993), 287294.##[18] S. Y. Chou, L. Q. Dat and F. Y. Vincent, A revised method for ranking fuzzy numbers using##maximizing set and minimizing set, Comput. Ind. Eng., 61 (2011), 13421384.##[19] T. Chu and C. Tsao, Ranking fuzzy numbers with an area between the centroid point and##orginal point, Comput. Math. Appl., 43 (2002), 11117.##[20] L. Q. Dat, F. Y. Vincent and S. Y chou, An improved ranking method for fuzzy numbers##based on the centroidindex, International Fuzzy Systems, 14 (3) (2012), 413419.##[21] K. Deep, M. L. Kansal and K. P. Singh, Ranking of alternatives in fuzzy environment using##integral value, J. Math. Stat. Allied Fields, 1(2) (2007), 20702077.##[22] M. Delgado, M. A. Vila and W. Voxman, On a canonical representation of fuzzy numbers,##Fuzzy Sets and Systems, 93 (1998), 125135.##[23] Y. Deng and Q. Liu, A TOPSISbased centroid index ranking method of fuzzy numbers and##its application in decisionmaking, Cybernetic and Systems, 36 (2005), 581595.##[24] Y. Deng, Z. F. Zhu and Q. Liu, Ranking fuzzy numbers with an area method using of gyration,##Comput. Math. Appl., 51 (2006), 11271136.##[25] D. Dubios and H. Prade, Operations on fuzzy numbers, Internat. J. System Sci., 9 (1978),##[26] M. S. Garcia and M. T. Lamata, A modication of the index of Liou and Wang for ranking##fuzzy numbers, Int.J. Uncer. Fuzz. Know. Based Syst., 14(4) (2007).##[27] T. Hajjari, On deviation degree methods for ranking fuzzy numbers, Australian Journal of##Basic and Applied Sciences, 5(5) (2011), 750758.##[28] T. Hajjari, Ranking of fuzzy numbers based on ambiguity degree, Australian Journal of Basic##and Applied Sciences., 5(1) (2011), 6269. ##[29] T. Hajjari and S. Abbasbandy, A note on " The revised method of ranking LR fuzzy number##based on deviation degree", Expert Syst with Applications, 38 (2011), 1349113492.##[30] R. Jain, Decisionmaking in the presence of fuzzy variable, IEEE Trans. Systems Man and##Cybernet., 6 (1976), 698703.##[31] R. Jain, A procedure for multiaspect decision making using fuzzy sets. Internat. J. Systems##Sci., 8 (1977), 17.##[32] A. Kumar, P. Singh, P. Kaur and A. Kaur, RM approach for ranking of LR type generalized##fuzzy numbers, Soft Cumput, 15 (2011), 13731381.##[33] A. Kumar, P. Singh and A. Kuar, Ranking of generalized exponentialfuzzy numbers using##integral value approach, Int.J.Adv.Soft.Comput.Appl., 2(2) (2010), 221230.##[34] A. Kumar, P. Singh, P. Kuar and A. Kuar, A new approach for ranking of L R type##generalized fuzzy numbers, Expert Syst. Appl., 38 (2011), 1090610910.##[35] A. Kumar, P. Singh, A. Kaur and P. Kaur, A new approach for ranking of nonnormal pnorm##trapezoidal fuzzy numbers, Comput. Math. Appl., 57 (2011), 881887.##[36] T. S. Liou and M. J. Wang, Ranking fuzzy numbers with integral value, Fuzzy Sets and##Systems, 50 (1992), 247255.##[37] X. W. Liu and S. L. Han, Ranking fuzzy numbers with preference weighting function expec##tationc, Comput. Math. Appl., 49 (2005), 14551465.##[38] S. Murakami, H. Maeda and S. Imamura, Fuzzy decision analysis on development of cen##tralized regional energycontrol system, Proceeding of the IFAC Symposium Marseille, (1983),##[39] P. Phani Bushan Rao and R. Shankar, Ranking fuzzy numbers with a distance method##using circumcenter of centroids an index of modality, Advance in Fuzzy Systems,##dio:10.1155/2011/178308, 2011.##[40] S. Rezvani, Ranking generalized fuzzy numbers with Euclidian distance by the incentre of##centroid, Mathematica Aeterna, 3 (2013), 103114.##[41] F. Y. Vincent and L. Q. Dat, An improved ranking method for fuzzy numbers with integral##values, Appl. Soft Comput., 14 (2014), 603608.##[42] Y. J.Wang and H. Sh. Lee, The revised method of ranking fuzzy numbers with an erea between##the centroid and original points, Comput. Math. Appl., 55 (2008), 20332042.##[43] Z. X. Wang, Y. J. Liu, Z. P. Fan and B. Feng, Ranking LR fuzzy number based on diviation##degree, Information Sciences, 179 (2009), 20702077.##[44] W. Wang and Z. Wang, Total orderings dened on the set of all fuzzy numbers, Fuzzy Sets##and Sysemts, 243 (2014), 131141.##[45] Y. M. Wang and Y. Luo, Area ranking of fuzzy numbers based on positive and negative ideal##points, Comput. Math. Appl., 58 (2009), 17761779.##[46] X. Wang and E. E. Kerre, Reasonable properties for the ordering of fuzzy quantities (I),##Fuzzy Sets and Systems, 118 (2001), 375385.##[47] X. Wang and E. E. Kerre, Reasonable properties for the ordering of fuzzy quantities (II),##Fuzzy Sets and Systems, 118 (2001), 387405.##[48] R. R. Yager, On choosing between fuzzy subsets, Kybernetes, 9 (1980), 151154.##[49] R. R. Yager, On a general class of fuzzy connective, Fuzzy Sets and Systems, 4 (1980),##[50] R. R. Yager, A procedure for ordering fuzzy subests of the unit interval, Inform. Sciences, 24##(1981), 143161.##[51] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338353.##]
Fixed Fuzzy Points of Fuzzy Mappings in Hausdorff Fuzzy Metric Spaces with Application
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Recently, Phiangsungnoen et al. [J. Inequal. Appl. 2014:201 (2014)] studied fuzzy mappings in the framework of Hausdorff fuzzy metric spaces.Following this direction of research, we establish the existence of fixed fuzzy points of fuzzy mappings. An example is given to support the result proved herein; we also present a coincidence and common fuzzy point result. Finally, as an application of our results, we investigate the existence of solution for somerecurrence relations associated to the analysis of quicksort algorithms.
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Calogero
Vetro
Dipartimento di Matematica e Informatica, Universita degli Studi
di Palermo, Via Archirafi 34, 90123 Palermo, Italy
Dipartimento di Matematica e Informatica,
Italy
calogero.vetro@unipa.it


Mujahid
Abbas
Department of Mathematics and Applied Mathematics, University of
Pretoria, Lynnwood road, Pretoria 0002, South Africa, Department of Mathematics,
Syed Babar Ali School of Science and Engineering (SBASSE), Lahore University of
Management Sciences
Department of Mathematics and Applied Mathematics,
Pakistan
mujahid.abbas@up.ac.za


Basit
Ali
Department of Mathematics, Syed Babar Ali School of Science and Engi
neering (SBASSE), Lahore University of Management Sciences (LUMS), Lahore, 54792,
Pakistan
Department of Mathematics, Syed Babar Ali
Pakistan
basit.aa@gmail.com
Fuzzy metric space
Fuzzy mapping
Fixed fuzzy point
Quicksort algorithm
[[1] B. Ali and M. Abbas, Suzuki type xed point theorem for Fuzzy mappings in ordered metric##spaces, Fixed Point Theory Appl., 2013:9 (2013), 119.##[2] J. W. de Bakker and E. P. de Vink, A metric approach to control ##ow semantics, in: Proc.##Eleventh Summer Conference on General Topology and Applications, Ann. New York Acad.##Sci., 806 (1996), 1127.##[3] J. W. de Bakker and E. P. de Vink, Denotational models for programming languages: appli##cations of Banach's xed point theorem, Topology Appl., 85(13) (1998), 3552.##[4] J. W. de Bakker and E. P. de Vink, Control Flow Semantics, Cambridge, MA, USA: The##MIT Press, 1996.##[5] A. Deb Ray and P. K. Saha, Fixed point theorems on generalized fuzzy metric spaces,. Hacet.##J. Math. Stat., 39(1) (2010), 19.##[6] V. D. Estruch and A. Vidal, A note on xed fuzzy points for fuzzy mappings, Rend Istit.##Univ. Trieste, 32 (2001), 3945.##[7] J. X. Fang, On xed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems, 46(1)##(1992), 107113.##[8] P. Flajolet, Analytic analysis of algorithms, In: W. Kuich (Ed.), Automata, Languages and##Programming, 19th Internat. Colloq. ICALP'92, Vienna, July 1992, in: Lecture Notes in##Computer Science, Berlin: Springer, 623 (1992), 186210.##[9] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy sets and Systems,##64(3) (1994), 395399.##[10] A. George and P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets##and Systems, 90(3) (1997), 365368.##[11] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27(3) (1988),##[12] V. Gregori and S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets and Sys##tems, 115(1) (2000), 485489.##[13] V. Gregori and A. Sapena, On xed point theorems in fuzzy metric spaces, Fuzzy Sets and##Systems, 125(2) (2002), 245253.##[14] R. H. Haghi, Sh. Rezapour and N. Shahzad, Some xed point generalizations are not real##generalizations, Nonlinear Anal., 74(5) (2011), 17991803.##[15] S. Heilpern, Fuzzy mappings and xed point theorem, J. Math. Anal. Appl., 83(2) (1981),##[16] G. Kahn, The semantics of a simple language for parallel processing, in: Proc. IFIP Congress,##NorthHolland, Amsterdam: Elsevier, (1974), 471475.##[17] F. Kiany and A. AminiHarandi, Fixed point and endpoint theorems for setvalued fuzzy##contraction maps in fuzzy metric spaces, Fixed Point Theory Appl., 2011:94 (2011), 19.##[18] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11(5)##(1975), 336344.##[19] R. L. Kruse, Data structures and program design, PrenticeHall, Inc., Upper Saddle River,##NJ, USA, 1984. ##[20] Y. Liu and Z. Li, Coincidence point theorems in probabilistic and fuzzy metric spaces, Fuzzy##Sets and Systems, 158(1) (2007), 5870.##[21] S. G. Matthews, Partial metric topology, in: Proc. 8th Summer Conference on General Topol##ogy and Applications, Ann. New York Acad. Sci., 728 (1994), 183197.##[22] D. Mihet, On the existence and the uniqueness of xed points of Sehgal contractions, Fuzzy##Sets and Systems, 156(1) (2005), 135141.##[23] D. Mihet, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets and Systems,##158(8) (2007), 915921.##[24] N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings on complete##metric spaces, J. Math. Anal. Appl., 141(1) (1989), 177188.##[25] S. B. Nadler, Multivalued contraction mappings, Pacic J. Math., 30(2) (1969), 475488.##[26] S. Phiangsungnoen, W. Sintunavarat and P. Kumam, Fuzzy xed point theorems in Hausdor##fuzzy metric spaces, J. Inequal. Appl., 2014:201 (2014), 110.##[27] A. Razani, A contraction theorem in fuzzy metric space, Fixed Point Theory Appl., 2005(3)##(2005), 257265.##[28] J. RodrguezLopez and S. Romaguera, The Hausdor fuzzy metric on compact sets, Fuzzy##Sets and Systems, 147(2) (2004), 273283.##[29] S. Romaguera, A. Sapena and P. Tirado, The Banach xed point theorem in fuzzy quasi##metric spaces with application to the domain of words, Topology Appl., 15(10) (2007),##21962203.##[30] R. Saadati, S. M. Vaezpour and Y. J. Cho, Quicksort algorithm: Application of a xed point##theorem in intuitionistic fuzzy quasimetric spaces at a domain of words, J. Comput. Appl.##Math., 228(1) (2009), 219225.##[31] P. Salimi, C. Vetro and P. Vetro, Some new xed point results in nonArchimedean fuzzy##metric spaces, Nonlinear Anal. Model. Control, 18(3) (2013), 344358.##[32] B. Schweizer and A. Sklar, Statistical metric spaces, Pacic J. Math., 10(1) (1960), 385389.##[33] C. S. Sen, Fixed degree for fuzzy mappings and a generalization of Ky Fan's theorem, Fuzzy##Sets and Systems, 24(1) (1987), 103112.##[34] T. Som and R. N. Mukherjee, Some xed point theorems for fuzzy mappings, Fuzzy Sets and##Systems, 33(2) (1989), 213219.##[35] D. Turkoglu and B. E. Rhoades, A xed fuzzy point for fuzzy mapping in complete metric##spaces, Math. Commun., 10(2) (2005), 115121.##[36] L. A. Zadeh, Fuzzy Sets, Inf. Control, 8(3) (1965), 338353.##]
Some classes of statistically convergent sequences of fuzzy numbers generated by a modulus function
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The purpose of this paper is to generalize the concepts of statisticalconvergence of sequences of fuzzy numbers defined by a modulus functionusing difference operator $Delta$ and give some inclusion relations.
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U.
Cakan
Department of Mathematics, Nevsehir Hac Bektas Veli University, Nevsehir
Turkey
Department of Mathematics, Nevsehir Hac Bektas
Turkey
umitcakan@gmail.com


Y.
Altin
Department of Mathematics, Firat University, ElazigTurkey
Department of Mathematics, Firat University,
Turkey
yaltin23@yahoo.com
Sequence of fuzzy numbers
Statistical convergence
Modulus function
[[1] H. Altnok, R. C olak and M. Et, dierence sequence spaces of fuzzy numbers, Fuzzy Sets##and Systems, 160(21) (2009), 3128{3139.##[2] H. Altnok and R. C olak, Almost lacunary statistical and strongly almost lacunary conver##gence of generalized dierence sequences of fuzzy numbers, J. Fuzzy Math., 17(4) (2009),##[3] H. Altnok and M. Mursaleen, Statistical boundedness for sequences of fuzzy numbers,##Taiwanese Journal of Mathematics, 15(5) (2011), 20812093.##[4] M. Basarr and M. Mursaleen, Some sequence spaces of fuzzy numbers generated by innite##matrices, J. Fuzzy Math., 11(3) (2003), 757764.##[5] J. Connor, A topological and functional analytic approach to statistical convergence, Analysis##of divergence (Orono, ME, 1997), 403{413, Appl. Numer. Harmon. Anal., Birkhauser Boston,##Boston, MA, 1999.##[6] P. Diamond and P. Kloeden, Metric spaces of fuzzy sets, Fuzzy Sets and Systems, 35 (1990),##[7] M. Et and R. C olak, On some generalized dierence sequence spaces, Soochow J. Math.,##21(4) (1995), 377386.##[8] M. Et, H. Altnok and R. C olak, On statistical convergence of dierence sequences of##fuzzy numbers, Inform. Sci., 176(15) (2006), 2268{2278.##[9] H. Fast, Sur la convergence statistique, Colloq. Math., (1951), 241244.##[10] J. A. Fridy, On statistical convergence, Analysis., 5 (1985), 301313.##[11] H. Kzmaz, On certain sequence spaces, Canadian Math. Bull., 24 (1981), 169176.##[12] J. S. Kwon, On statistical and pCesaro convergence of fuzzy numbers, Korean J. Comput.##Appl. Math., 7(1) (2000), 195203.##[13] M. Matloka, Sequences of fuzzy numbers, BUSEFAL, 28 (1986), 2837. ##[14] M. Mursaleen and M. Basarr, On some new sequence spaces of fuzzy numbers, Indian J.##Pure and Appl. Math., 34(9) (2003), 1351{1357.##[15] S. Nanda, On sequences of fuzzy numbers, Fuzzy Sets and Systems, 33 (1989), 123126.##[16] H. Nakano, Concave modulars, J. Math. Soc. Japan, 5 (1953), 29{49.##[17] F. Nuray and E. Savas, Statistical convergence of fuzzy numbers, Math. Slovaca, 45(3) (1995),##[18] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30 (1980),##[19] B. Sarma, On a class of sequences of fuzzy numbers dened by modulus function, International##Journal of Science & Technology, 2(1) (2007), 2528.##[20] I. J. Schoenberg, The integrability of certain functions and related summability methods,##Amer. Math. Monthly, 66 (1959), 361375.##[21] O. Talo and F. Basar, Certain spaces of sequences of fuzzy numbers dened by a modulus##function, Demonstratio Math., 43(1) (2010), 139{149.##[22] B. C. Tripathy and A. J. Dutta, Bounded variation double sequence space of fuzzy real##numbers, Comput. Math. Appl., 59(2) (2010), 1031{1037.##[23] L. A. Zadeh, Fuzzy sets, Inform and Control, 8 (1965), 338353.##]
Categoricallyalgebraic topology and its applications
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This paper introduces a new approach to topology, based on category theory and universal algebra, and called categoricallyalgebraic (catalg) topology. It incorporates the most important settings of latticevalued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$fuzzy topology on $L$fuzzy sets of C.~Guido. Moreover, its respective categories of topological structures are topological over their ground categories. The theory also extends the notion of topological system of S.~Vickers (and its numerous manyvalued modifications of J.~T.~Denniston, A.~Melton and S.~E.~Rodabaugh), and shows that the categories of catalg topological structures are isomorphic to coreflective subcategories of the categories of catalg topological systems. This extension initiates a new approach to soft topology, induced by the concept of soft set of D.~Molodtsov, and currently pursued by various researchers.
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Sergey A.
Solovyov
Institute of Mathematics, Faculty of Mechanical Engineering,
Brno University of Technology, Technicka 2896/2, 616 69 Brno, Czech Republic and
Institute of Mathematics and Computer Science, University of Latvia, Raina bulvaris
29, LV1459 Riga,
Institute of Mathematics, Faculty of Mechanical
Latvia
sergejs.solovjovs@lumii.lv
Categoricallyalgebraic topology
Latticevalued topology
Soft topology
Topological category
Topological system
Topological theory
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Convergence, Consistency and Stability in Fuzzy Differential Equations
2
2
In this paper, we consider Firstorder fuzzy differential equations with initial value conditions. The convergence, consistency and stability of difference method for approximating the solution of fuzzy differential equations involving generalized Hdifferentiability, are studied. Then the local truncation error is defined and sufficient conditions for convergence, consistency and stability of difference method are provided and fuzzy stiff differential equation and one example are presented to illustrate the accuracy and capability of our proposed concepts.
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95
112


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


K.
Maleknejad
Department of Mathematics, Karaj Branch, Islamic Azad University,
Karaj, Iran
Department of Mathematics, Karaj Branch,
Iran
maleknejad@iust.ac.ir


S.
Khezerloo
Department of Mathematics, Islamic Azad University  South Tehran
Branch, Tehran, Iran
Department of Mathematics, Islamic Azad University
Iran
s_khezerloo@azad.ac.ir


M.
Khezerloo
Department of Mathematics, Institute for Advanced Studies in Basic
Sciences(IASBS), P.O. BOX 451951159, Zanjan, Iran
Department of Mathematics, Institute for
Iran
khezerloo@iasbs.ac.ir
Consistence
Stability
Local truncation error
Generalized differentiability
Fuzzy stiff differential equation
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Interval Type2 Fuzzy Rough Sets and Interval Type2 Fuzzy Closure Spaces
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2
The purpose of the present work is to establish a onetoone correspondence between the family of interval type2 fuzzy reflexive/tolerance approximation spaces and the family of interval type2 fuzzy closure spaces.
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113
125


Shambhu
Sharan
School of Advanced Sciences, VIT University,Vellore632014,Tamil
Nadu, India
School of Advanced Sciences, VIT University,Vellor
India
shambhupuremaths@gmail.com


S. P.
Tiwari
Department of Applied Mathematics, Indian School of Mines, Dhanbad
826004, India
Department of Applied Mathematics, Indian
India
sptiwarimaths@gmail.com


V. K.
Yadav
Department of Applied Mathematics, Indian School of Mines, Dhanbad
826004, India
Department of Applied Mathematics, Indian
India
vijayyadav3254@gmail.com
Interval type2 fuzzy set
Interval type2 fuzzy rough set
Interval type2 fuzzy reflexive approximation space
Interval type2 fuzzy tolerance approximation space
Interval type2 fuzzy closure space
Interval type2 fuzzy topology
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Boundedness of linear orderhomomorphisms in $L$topological vector spaces
2
2
A new definition of boundedness of linear orderhomomorphisms (LOH)in $L$topological vector spaces is proposed. The new definition iscompared with the previous one given by Fang [The continuity offuzzy linear orderhomomorphism, J. Fuzzy Math. 5 (4) (1997)829$$838]. In addition, the relationship between boundedness andcontinuity of LOHs is discussed. Finally, a new uniform boundednessprinciple in $L$topological vector spaces is established in thesense of a new definition of uniform boundedness for a family ofLOHs.
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127
135


HuaPeng
Zhang
School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
School of Science, Nanjing University of
China
huapengzhang@163.com


JinXuan
Fang
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
School of Mathematical Sciences, Nanjing
China
jxfang@njnu.edu.cn
$L$topological vector space
Linear orderhomomorphism
Boundedness
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Distinct Fuzzy Subgroups of a Dihedral Group of Order $2pqrs$ for Distinct Primes $p, , q, , r$ and $s$
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In this paper we classify fuzzy subgroups of the dihedral group $D_{pqrs}$ for distinct primes $p$, $q$, $r$ and $s$. This follows similar work we have done on distinct fuzzy subgroups of some dihedral groups.We present formulae for the number of (i) distinct maximal chains of subgroups, (ii) distinct fuzzy subgroups and (iii) nonisomorphic classes of fuzzy subgroups under our chosen equivalence and isomorphism. Some results presented here hold for any dihedral group of order $2n$ where $n$ is a product of any number of distinct primes.
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Babington
Makamba
Department of Mathematics, University of Fort Hare, Alice
5700 , Eastern Cape , South Africa
Department of Mathematics, University of
South Africa
bmakamba@ufh.ac.za


Odilo
Ndiweni
Department of Mathematics, University of Fort Hare, Alice 5700 ,
Eastern Cape , South Africa
Department of Mathematics, University of
South Africa
ondiweni@ufh.ac.za
Dihedral group
Equivalence
Isomorphism
Fuzzy subgroup
Maximal chain
Keychain
Distinguishing factor
[[1] S. Branimir and A. Tepavcevic, A note on a natural equivalence relation on fuzzy power set,##Fuzzy Sets and Systems, 148(2) (2004), 201{210.##[2] C. Degang, J. Jiashang, W. Congxin and E. C. C. Tsang, Some notes on equivalent fuzzy##sets and fuzzy subgroups, Fuzzy Sets and systems, 152(2) (2005), 403{409.##[3] V. Murali and B. B. Makamba, On an equivalence of fuzzy subgroups I, Fuzzy Sets and##Systems 123(2) (2001), 259{264.##[4] O. Ndiweni and B. B. Makamba, Classication of fuzzy subgroups of a dihedral group of##order 2pqr for distinct primes p, q and r, International Jounal of Mathematical Sciences and##Engineering Applications, 6(4) (2012) , 159{174.##[5] M. Pruszyriska and M. Dudzicz, On isomorphism between nite chains, Journal of Formalised##Mathematics, 12(1) (2003) , 1{2.##[6] S. Ray, Isomorphic fuzzy groups, Fuzzy Sets and Systems, 50(2) (1992) , 201{207.##[7] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971) , 512{517.##[8] M. Tarnauceanu and L. Bentea, On the number of subgroups of nite abelian groups, Fuzzy##Sets and Systems, 159(10) (2008) , 1084{1096.##[9] A. C. Volf, Counting fuzzy subgroups and chains of subgroups, Fuzzy Systems and Articial##Intelligence, 10(3) (2004) , 191{200.##]
Solvable $L$subgroup of an $L$group
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In this paper, we study the notion of solvable $L$subgroup of an $L$group and provide its level subset characterization and this justifies the suitability of this extension. Throughout this work, we have used normality of an $L$subgroup of an $L$group in the sense of Wu rather than Liu.
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166


Iffat
Jahan
Department of Mathematics, Ramjas College,, University of Delhi,,
Delhi110007, India
Department of Mathematics, Ramjas College,,
India
ij.umar@yahoo.com


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
$L$algebra
$L$subgroup
Normal $L$subgroup
Solvable $L$subgroup
Derived series
Solvable series
[[1] N. Ajmal, Fuzzy groups with sup property, Inform. Sci., 93 (1996), 247264.##[2] N. Ajmal and I. Jahan , A study of normal fuzzy subgroups and characteristic fuzzy subgroups##of a fuzzy group, Fuzzy Information and Engineering, 2 (2012), 123143.##[3] N. Ajmal and I. Jahan, Nilpotency and theory of Lsubgroups of an Lgroup, Fuzzy Informa##tion and Engineering, 6 (2014), 117.##[4] 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 (2009), 40704082.##[5] N. Ajmal and I. Jahan,Generated Lsubgroup of an Lgroup, Iranian Journal of Fuzzy Sys##tems, 12(2) (2015), 129136.##[6] J. A. Goguen, Lfuzzy sets, J. Math. Anal. Appl., 18 (1967), 145174.##[7] K. C. Gupta and B. K. Sarma, nilpotent fuzzy groups, Fuzzy Sets and Systems, 101 (1999),##[8] W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems, 8 (1982),##[9] D. S. Malik, J. N. Mordeson and P. S. Nair, Fuzzy normal subgroups in fuzzy subgroups, J.##Korean Math. Soc., 29 (1992), 18.##[10] L. Martinez, L Fuzzy subgroups of fuzzy groups and fuzzy ideals of fuzzy rings, J. Fuzzy##Math., 3 (1995), 833849.##[11] J. N. Mordeson, K. R. Bhutani and A. Rosenfeld, Fuzzy group theory, Springer, 2005.##[12] J. N. Mordeson and D. S. Malik, Fuzzy commutative algebra, World Scientic, 1998.##[13] N. P. Mukherjee and P. Bhattacharya, Fuzzy groups: some grouptheoretic analogs, Inform.##Science, 39 (1986), 247268.##[14] A. S. Prajapati and N. Ajmal, Maximal ideals of Lsubrings, J. Fuzzy Math., 15 (1999),##[15] A. S. Prajapati and N. Ajmal,Maximal ideals of L{subrings. II, J. Fuzzy Math., 15 (2007),##[16] M. G. Ranitovic and A. Tepavcevic, General form of latticevalued fuzzy sets under the##cutworthy approach, Fuzzy Sets and Systems, 158 (2007), 12131216.##[17] S. Ray, Solvable fuzzy groups, Inform. Science 75 (1993) 4761.##[18] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512517.##[19] B. K. Sarma, Solvable fuzzy groups, Fuzzy Sets and Systems, 106 (1999), 463467.##[20] B. Seselja, D. Stojic and A. Tepavcevic, On existence of Pvalued fuzzy sets with a given##collection of cuts, Fuzzy Sets and Systems, 161 (2010), 763768.##[21] B. Seselja and A. Tepavcevic,Completion of ordered structures by cuts of fuzzy sets: an##overview, Fuzzy Sets and Systems, 136 (2003), 119.##[22] W. Wu, Normal fuzzy subgroups, Fuzzy Math., 1 (1981), 2130.##]
Persiantranslation vol. 12, no.3, June 2015
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