2013
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ON FUZZY NEIGHBORHOOD BASED CLUSTERING
ALGORITHM WITH LOW COMPLEXITY
ON FUZZY NEIGHBORHOOD BASED CLUSTERING
ALGORITHM WITH LOW COMPLEXITY
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The main purpose of this paper is to achieve improvement in thespeed of Fuzzy Joint Points (FJP) algorithm. Since FJP approach is a basisfor fuzzy neighborhood based clustering algorithms such as NoiseRobust FJP(NRFJP) and Fuzzy Neighborhood DBSCAN (FNDBSCAN), improving FJPalgorithm would an important achievement in terms of these FJPbased methods. Although FJP has many advantages such as robustness, auto detectionof the optimal number of clusters by using cluster validity, independency fromscale, etc., it is a little bit slow. In order to eliminate this disadvantage, by improving the FJP algorithm, we propose a novel Modied FJP algorithm, whichtheoretically runs approximately n= log2 n times faster and which is less complex than the FJP algorithm. We evaluated the performance of the ModiedFJP algorithm both analytically and experimentally.
1
The main purpose of this paper is to achieve improvement in thespeed of Fuzzy Joint Points (FJP) algorithm. Since FJP approach is a basisfor fuzzy neighborhood based clustering algorithms such as NoiseRobust FJP(NRFJP) and Fuzzy Neighborhood DBSCAN (FNDBSCAN), improving FJPalgorithm would an important achievement in terms of these FJPbased methods. Although FJP has many advantages such as robustness, auto detectionof the optimal number of clusters by using cluster validity, independency fromscale, etc., it is a little bit slow. In order to eliminate this disadvantage, by improving the FJP algorithm, we propose a novel Modied FJP algorithm, whichtheoretically runs approximately n= log2 n times faster and which is less complex than the FJP algorithm. We evaluated the performance of the ModiedFJP algorithm both analytically and experimentally.
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20
Gozde
Ulutagay
Gozde
Ulutagay
Department of Industrial Engineering, Izmir University, Gursel
Aksel Blv 14, Uckuyular, Izmir, Turkey
Department of Industrial Engineering, Izmir
Turkey
gozde.ulutagay@izmir.edu.tr
Efendi
Nasibov
Efendi
Nasibov
Department of Computer Science, Dokuz Eylul University, Izmir,
35160, Turkey, Institute of Cybernetics, Azerbaijan National Academy of Sciences,
Azerbaijan
Department of Computer Science, Dokuz Eylul
Turkey
efendi nasibov@yahoo.com
Clustering
Fuzzy neighborhood relation
Complexity
Modied FJP
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Chehreghani, Improving densitybased methods##for hierarchical clustering of web pages, Data & Knowledge Engineering, 67 (2008), 30{50.##[6] T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to algorithms, The##MIT Press, 2001.##[7] A. P. Dempster, N. M. Laird and D. B. Rubin, Maximum likelihood from incomplete data##via the EM algorithm, Journal of Royal Statistical Society, Series B, 39 (1977), 1{38.##[8] J. C. Dunn, A fuzzy relative of the ISODATA process and its use in detecting compact wellseparated##clusters, Journal of Cybernetics, 3(3) (1973), 32{57.##[9] M. Ester, H. P. Kriegel, J. Sander and X. Xu, A densitybased algorithm for discovering clusters##in large spatial databases with noise, In Proc. 2nd International Conference on Knowledge##Discovery and Data Mining, (1996), 226{231.##[10] D. Fisher, Knowledge acquisition via conceptual clustering, Machine Learning, 2 (1987),##[11] S. Guha, R. Rastogi and K. Shim, CURE: an efficient clustering algorithms for large##databases, In: Proceeding ACM SIGMOD International Conference on Management of Data,##Seattle, WA, (1998), 73{84.##[12] J. Han and M. Kamber, Data mining concepts and techniques, Morgan Kaufmann Publishers,##San Francisco, CA, 2001.##[13] A. Hinneburg and A. K. Daniel, An efficient approach to clustering in large multimedia##databases with noise, Proceedings of the 4th Int. Conference on Knowledge Discovery and##Data Mining (KDD98), New York, (1998), 58{65.##[14] P. Hore, L. O. Hall, D. B. Goldgof, Y. GU, A. A. Maudsley and A. Darkazanli, A scalable##framework for segmenting magnetic resonance images, Journal of Signal Processing Systems,##54 (2009), 183{203.##[15] E. Januzaj, H. P. Kriegel and M. Pfei##e, DBDC: density based distributed clustering, 5th##International Conference on Extending Database Technology (EDBT), Heraklion, Greece,##(2004a), 88{105.##[16] E. Januzaj, H. P. Kriegel and M. Pfei##e, Scalable density based distributed clustering, 15th##International Conference on Machine Learning (ECML) and the 8th European Conference on##Principles and Practice of Knowledge Discovery in Databases (PKDD), Pisa, Italy, 2004b.##[17] G. Karypis, E. H. Han and V. Kumar, CHAMELEON: a hierarchical clustering algorithm##using dynamic modeling, IEEE Computer, 32(8) (1999), 68{75.##[18] L. Kaufman and P. J. Rousseuw, Finding groups in data: an introduction to cluster analysis,##John Wiley&Sons, Inc, 1990.##[19] E. Mehdizadeh, S. SadiNezhad and R. TavakkoliMoghaddam, Optimization of fuzzy clustering##criteria by a hybrid PSO and fuzzy cmeans clustering algorithm, Iranian Journal of##Fuzzy Systems, 5(3) (2008), 1{14.##[20] E. N. Nasibov and G. Ulutagay, A new approach to clustering problem using the fuzzy joint##points method, Automatic Control and Computer Sciences, 39(6) (2005), 8{17.##[21] E. N. Nasibov and G. Ulutagay, On the fuzzy joint points method for fuzzy clustering problem,##Automatic Control and Computer Sciences, 40(5) (2006), 33{44.##[22] E. N. Nasibov and G. Ulutagay, A new unsupervised approach for fuzzy clustering, Fuzzy##Sets and Systems, 158(19) (2007), 2118{2133.##[23] E. N. Nasibov and G. Ulutagay, Robustness of densitybased clustering methods with various##neighborhood relations, Fuzzy Sets and Systems, 160(24) (2009), 3601{3615.##[24] T. R. Ng and J. Han, Efficient and effective clustering methods for spatial data mining,##Proceedings of the 20th Very Large Databases Conference (VLDB94), Santiago, Chile, (1994),##[25] J. K. Parker, L. O. Hall and A. Kandel, Scalable fuzzy neighborhood DBSCAN, IEEE Interna##tional Conference on Fuzzy Systems, Barcelona, Spain, doi: 10.1109/FUZZY.2010.5584527,##(2010), 1{8.##[26] W. Pedrycz and F. Gomide, An introduction to fuzzy sets, MIT Press, MA, 1998. ##[27] W. Pedrycz, Distributed and collaborative fuzzy modeling, Iranian Journal of Fuzzy Systems,##4(1) (2007), 1{19.##[28] J. Sander, M. Ester, H. P. Kriegel and X. Xu,Densitybased clustering in spatial databases:##the algorithm GDBSCAN and its applications, Data Mining and Knowledge Discovery, 2##(1998), 169{194.##[29] G. Sheikholeslami, S. Chatterjee and A. Zhang, WaveCluster: a multiresolution clustering##approach for very large spatial databases, Proceedings of the 24th Very Large Databases##Conference (VLDB 98), New York, 1998.##[30] G. Ulutagay and E. Nasibov, Fuzzy and crisp clustering methods based on the neighborhood##concept: a comprehensive review, Journal of Intelligent and Fuzzy Systems, 23 (2012), 271{##[31] W. Wang, Y. Jiong and R. Muntz, STING: a statistical information grid approach to spatial##data mining, Proceedings of the 23rd Very Large Databases Conference (VLDB 1997), Athens,##Greece, 1997.##[32] X. Xiaowei, E. Martin, H. P. Kriegel and J. Sander, A distributionbased clustering algorithm##for mining in large spatial databases, Proceedings of the 14th Int. Conference on Data##Engineering (ICDE98), Orlando, Florida, (1998), 324{331.##[33] A. Z. Xu, J. Chen and J. Wu, Clustering algorithm for intuitionistic fuzzy sets, Information##Sciences, 178 (2008), 3775{3790.##[34] R. R. Yager and D. P. Filev, Approximate clustering via the mountain method, IEEE Trans##actions on Systems, Man and Cybernetics, 24(8) (1994) , 1279{1284.##[35] X. L. Yang, Q. Song and Y. L. Wu, A robust deterministic algorithm for data clustering,##Data & Knowledge Engineering, 62 (2007), 84{100.##[36] T. Zhang, R. Ramakrishnan and M. Livny, BIRCH: an efficient data clustering method for##very large databases, Proceedings of the 1996 ACM SIGMOD Int. Conference on Management##of Data, Montreal, Canada, (1996), 103{113.##]
OPTIMAL CONTROL OF FUZZY LINEAR CONTROLLED
SYSTEM WITH FUZZY INITIAL CONDITIONS
OPTIMAL CONTROL OF FUZZY LINEAR CONTROLLED
SYSTEM WITH FUZZY INITIAL CONDITIONS
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2
In this article we found the solution of fuzzy linear controlled systemwith fuzzy initial conditions by using cuts and presentation of numbersin a more compact form by moving to the eld of complex numbers. Next, afuzzy optimal control problem for a fuzzy system is considered to optimize theexpected value of a fuzzy objective function. Based on Pontryagin MaximumPrinciple, a constructive equation for the problem is presented. In the lastsection, three examples are used to show that the method in eective to solvefuzzy and fuzzy optimal linear controlled systems.
1
In this article we found the solution of fuzzy linear controlled systemwith fuzzy initial conditions by using cuts and presentation of numbersin a more compact form by moving to the eld of complex numbers. Next, afuzzy optimal control problem for a fuzzy system is considered to optimize theexpected value of a fuzzy objective function. Based on Pontryagin MaximumPrinciple, a constructive equation for the problem is presented. In the lastsection, three examples are used to show that the method in eective to solvefuzzy and fuzzy optimal linear controlled systems.
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35
Marzieh
Najariyan
Marzieh
Najariyan
Department of Applied Mathematics, Ferdowsi University of
Mashhad, Mashhad, Iran
Department of Applied Mathematics, Ferdowsi
Iran
marzieh.najariyan@gmail.com
Mohamad Hadi
Farahi
Mohamad Hadi
Farahi
Department of Applied Mathematics, Ferdowsi University of
Mashhad, Mashhad, Iran and The center of Excellence on Modelling and Control
Systems (CEMCS)
Department of Applied Mathematics, Ferdowsi
Iran
farahi@math.um.ac.ir
Fuzzy linear controlled system
Optimal fuzzy controlled system
PMP
[1. T. Allahviranloo and M. A. Kermani, Numerical methods for fuzzy linear partial dierential##equations under new denition for derivative, Iranian Journal of Fuzzy Systems, 7 (2010),##2. B. Bede, I. J. Rudas and A. L. Bencsik, First order linear fuzzy dierential equation sunder##generalized dierentiability, Information Science, 177 (2007), 16481662.##3. R. C. Dorf and R. H. Bishop, Modern control systems, Person Education, Inc. Upper Saddle##River, New Jersey, 07458 (2011).##4. P. Diamond and P. E. Kloeden, Metric space of Fuzzy sets, Theory And Applications, World##scientic publishing, 1994.##5. O. S. Fard and A. V. Kamyad, Modied kstep method for solving fuzzy initial value problems,##Iranian Journal of Fuzzy Systems, 8 (2011), 4963.##6. D. Filev and P. Angelove, Fuzzy optimal control, Fuzzy Sets and Systems, 47 (1992), 15156.##7. D. N. Georgiou, J. J. Nieto and R. RodriguezLopez, Initial value problems for higherorder##fuzzy dierential equations, Nonlinear Analysis, 63 (2005), 587600.##8. A. Khastan, J. J. Nieto and R. RodriguezLopez, Variation of constant formula for rst order##fuzzy dierential equations, Fuzzy Sets and Systems, 177 (2011), 2033.##9. A. Khastan and J. J. Nieto, A boundary value problem for second order fuzzy dierential##equations, Nonlinear Analysis, 72 (2010), 35833593.##10. J. J. Nieto, R. RodriguezLopez and M. VillanuevaPesqueira, Exact solution to the periodic##boundary value problem for a rstorder linear fuzzy dierential equation with impulses, Fuzzy##Optimization Decision Making, 10 (2011), 323339.##11. J. J. Nieto, A. Khastan and K. Ivaz, Numerical solution of fuzzy dierential equations under##generalized dierentiability, Nonlinear Analysis: Hybrid Systems, 3 (2009), 700707.##12. J. H. Park, J. S. Park and Y. C. Kwun, Controllability for the semilinear fuzzy integrodier##ential equations with nonlocal conditions, Lecture Notes in Articial Intelligence, LNAI 4223##(2006), 221230.##13. D. W. Pearson, A property of linear fuzzy dierential equations, Appl. Math. Lett., 10 (1997),##14. E. R. Pinch, Optimal control and the calculuse of variations, Oxford University Press Inc.,##New Yourk, 1995.##15. Z. Qin, Timehomogeneous fuzzy optimal control problems, http://www.orsc.edu.cn/process/##080415.pdf.##16. S. Ramezanzadeh and A. Heydari, Optimal control with fuzzy chance constraint, Iranian##Journal of fuzzy systems, 8 (2011), 3543.##17. S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (1987), 319330.##18. J. Xu, Z. Liao and J. J. Nieto, A class of dierential dynamical systems with fuzzy matrices,##Math. Anal. Appl., 368 (2010), 5468.##19. J. Xu, Z. Liao and Z. Hu, A class of linear dierential dynamical systems with fuzzy initial##condition, Fuzzy Sets and Systems, 158 (2007), 23392358.##20. Y. Zhu, A fuzzy optimal control model, Journal of uncertain systems, 3 (2009), 270279.##21. Y. Zhu, Fuzzy optimal control with application to portfolio selection, http://www.orsc.edu.cn/##process/080117.pdf.##]
$mathcal{I}_2$convergence of double sequences of\ fuzzy numbers
$mathcal{I}_2$convergence of double sequences of\ fuzzy numbers
2
2
In this paper, we introduce and study the concepts of $mathcal{I}_2$convergence, $mathcal{I}_2^{*}$convergence for double sequences of fuzzy real numbers, where $mathcal{I}_2$ denotes the ideal of subsets of $mathbb N times mathbb N$. Also, we study some properties and relations of them.
1
In this paper, we introduce and study the concepts of $mathcal{I}_2$convergence, $mathcal{I}_2^{*}$convergence for double sequences of fuzzy real numbers, where $mathcal{I}_2$ denotes the ideal of subsets of $mathbb N times mathbb N$. Also, we study some properties and relations of them.
37
50
Erdinc.
Dundar
Erdinc.
Dundar
Department of Mathematics, Afyon Kocatepe University, 03200
Afyonkarahisarn,Turkey
Department of Mathematics, Afyon Kocatepe
Turkey
erdincdundar79@gmail.com
Ozer
Talo
Ozer
Talo
Department of Mathematics, Celal Bayar University, 45040 Manisa,
Turkey
Department of Mathematics, Celal Bayar University,
Turkey
ozertalo@hotmail.com
Ideal
Double sequences
$mathcal{I}$Convergence
Fuzzy number sequences
[bibitem{BaFb}##B. Altay and F. Bad{s}ar, emph{Some new spaces of double sequences}, J. Math. Anal. Appl., textbf{309(1)} (2005), 7090.##bibitem{altýn}##H. Alt{i}nok, Y. Alt{i}n and M. Id{s}{i}k, emph{Statistical convergence and strong pCes'{a}ro summability of order $beta$ in sequences of fuzzy numbers}, Iranian Journal of Fuzzy##Systems, textbf{9(2)} (2012), 6373.##bibitem{bede}B. Bede and S. G. Gal, textit{Almost periodic fuzzynumbervalued functions},##Fuzzy Sets and Systems, textbf{147} (2004), 385403.##bibitem{ccba}##d{C}. CÇakan and B. Altay, emph{Statistically boundedness and statistical core##of double sequences}, J. Math. Anal. Appl., textbf{317} (2006), 690697.##bibitem{das 1}##P. Das, P. Kostyrko, W. Wilczy'{n}ski and P. Malik, emph{I and##$I^{*}$convergence of double sequences}, Math. Slovaca, textbf{58(5)} (2008), 605620.##bibitem{das 2}##P. Das and P. Malik, emph{On extremal Ilimit points of double##sequences}, Tatra Mt. Math. Publ., textbf{40} (2008), 91102.##bibitem{edba FU}##E. D"{u}ndar and B. Altay emph{$mathcal{I}_2$uniform convergence of double##sequences of functions}, (under communication).##bibitem{fang}J. X. Fang and H. Huang, textit{On the level convergence of a##sequence of fuzzy numbers}, Fuzzy Sets and Systems, textbf{147} (2004), 417415.##bibitem{fast}##H. Fast, emph {Sur la convergence statistique}, Colloq. Math.,##textbf{2} (1951), 241244.##bibitem{frst}##J. A. Fridy, emph{On statistical convergence}, Analysis,##textbf{5} (1985), 301313.##bibitem{fr c.o}##J. A. Fridy and C. Orhan, emph{Statistical limit superior and inferior}, Proc. Amer. Math. Soc., textbf{125} (1997), 36253631.##bibitem{frstlim}##J. A. Fridy, emph{Statistical limit points}, Proc. Amer. Math. Soc.,##textbf{118} (1993), 11871192.##bibitem{kos1}##P. Kostyrko, T. u{S}al'{a}t and W. Wilczy'{n}ski, emph{Iconvergence},##Real Anal. Exchange, textbf{26(2)} (2000), 669686.##bibitem{kos2}##P. Kostyrko, M. Mav{c}aj, T. u{S}al'{a}t and M. Sleziak, emph{Iconvergence##and extremal Ilimit points}, Math. Slovaca, textbf{55} (2005), 443464.##bibitem{kumar 1}##V. Kumar, emph{On I and $I^{*}$convergence of double sequences},##Math. Commun., textbf {12} (2007), 171181.##bibitem{kumar F}##V. Kumar and K. Kumar, emph{On the ideal convergence of sequences of fuzzy numbers}, Information Sciences, textbf{178} (2008), 46704678.##bibitem{Matloka}##M. Matloka, emph{Sequences of fuzzy numbers}, Busefal, textbf{28} (1986), 2837.##bibitem{mursest}##Mursaleen and O. H. H. Edely, emph{Statistical convergence of double##sequences}, J. Math. Anal. Appl., textbf{288} (2003), 223231.##bibitem{Nanda}##S. Nanda, emph{On sequences of fuzzy numbers}, Fuzzy Sets and Systems, textbf{33} (1989), 123126.##bibitem{nabiev}##A. Nabiev, S. Pehlivan and M. G"{u}rdal, emph{On ICauchy sequence},##Taiwanese J. Math., textbf {11(2)} (2007), 569576.##bibitem{nuray}##F. Nuray and W. H. Ruckle, emph{Generalized statistical convergence and convergence free spaces}, J. Math. Anal. Appl., textbf{245} (2000), 513527.##bibitem{nuray 2}##F. Nuray, emph{Lacunary statistical convergence of sequences of fuzzy numbers},##Fuzzy Sets and Systems, textbf{99} (1998), 353355.##bibitem{nuray 3}##F. Nuray and E. Savad{s}, emph{Statistical convergence of sequences of fuzzy numbers}, Math. Slovaca, textbf{45(3)} (1995), 269273.##bibitem{prinsgheim}##A. Pringsheim, emph{Zur theorie der zweifach unendlichen Zahlenfolgen},##Math. Ann., textbf{53} (1900), 289321.##bibitem{rath}##D. Rath and B. C. Tripaty, emph{On statistically convergence and##statistically Cauchy sequences}, Indian J. Pure Appl. Math., textbf{25(4)} (1994), 381386.##bibitem{saadati}##R. Saadati, emph{On the Ifuzzy topological spaces}, Chaos, Solitons and Fractals,##textbf{37} (2008), 14191426.##bibitem{salat st}##T. u{S}al'{a}t, emph {On statistically convergent sequences of##real numbers}, Math. Slovaca, textbf{30} (1980), 139150.##bibitem{Salat}##T. u{S}al'{a}t, B. C. Tripaty and M. Ziman, emph{On Iconvergence##field}, Ital. J. Pure Appl. Math., textbf {17} (2005), 4554.##bibitem{Savas1}##E. Savad{s}, emph{On statistical convergent sequences of fuzzy numbers},##Information Sciences, textbf{137} (2001), 277282.##bibitem{Savas2}##E. Savad{s} and Mursaleen, emph{On statistically convergent##double sequences of fuzzy numbers}, Information Sciences, textbf{162} (2004), 183192.##bibitem{Savas3}##E. Savad{s}, emph{A note on double sequences of fuzzy numbers}, Turk. Jour. Math., textbf{20(20)} (1996), 175178.##bibitem{Savas4}##E. Savad{s}, emph{$(A)_{Delta}$double sequence spaces of fuzzy numbers via orlicz function}, Iranian Journal of Fuzzy##Systems, textbf{8(2)} (2011), 91103.##bibitem{scho}##I. J. Schoenberg, emph {The integrability of certain functions and##related summability methods}, Amer. Math. Monthly, textbf {66}##(1959), 361375.##bibitem{otfb}##"{O}. Talo and F. Bad{s}ar, emph{Determination of the##duals of classical sets of sequences of fuzzy numbers and related##matrix transformations}, Comput. Math. Appl., textbf{58} (2009),##bibitem{tri 1}##B. Tripathy and B. C. Tripathy, emph{On Iconvergent double##sequences}, Soochow J. Math., textbf {31} (2005), 549560.##bibitem{tri 2}##B. C. Tripathy, emph{Statistically convergent double sequences}, Tamkang J. Math., textbf{34(3)} (2003), 231237.##bibitem{tri 3}##B. C. Tripathy and B. Sarma, emph{Double sequence spaces of fuzzy numbers defined by Orlicz function}, Acta Math. Sci., textbf{31B(1)} (2011), 134140.##bibitem{z}##L. A. Zadeh, textit{Fuzzy sets}, Information and Control, textbf{8}(1965), 338353.##]
ON APPROXIMATE CAUCHY EQUATION IN FELBIN'S TYPE
FUZZY NORMED LINEAR SPACES
ON APPROXIMATE CAUCHY EQUATION IN FELBIN'S TYPE
FUZZY NORMED LINEAR SPACES
2
2
n this paper we study the HyersUlamRassias stability of Cauchyequation in Felbin's type fuzzy normed linear spaces. As a resultwe give an example of a fuzzy normed linear space such that thefuzzy version of the stability problem remains true, while it failsto be correct in classical analysis. This shows how the category offuzzy normed linear spaces differs from the classical normed linearspaces in general.
1
n this paper we study the HyersUlamRassias stability of Cauchyequation in Felbin's type fuzzy normed linear spaces. As a resultwe give an example of a fuzzy normed linear space such that thefuzzy version of the stability problem remains true, while it failsto be correct in classical analysis. This shows how the category offuzzy normed linear spaces differs from the classical normed linearspaces in general.
51
63
I.
Sadeqi
I.
Sadeqi
Department of Mathematics, Sahand university of technology, Tabriz
Iran
Department of Mathematics, Sahand university
Iran
esadeqi@sut.ac.ir
F.
Moradlou
F.
Moradlou
Department of Mathematics, Sahand university of technology, Tabriz
Iran
Department of Mathematics, Sahand university
Iran
moradlou@sut.ac.ir
M.
Salehi
M.
Salehi
Department of Mathematics, Sahand university of technology, Tabriz
Iran
Department of Mathematics, Sahand university
Iran
mitpaydar@yahoo.com
Fuzzy real number
Fuzzy normed space
HyersUlamRassias stability
[[1] T. Bag and S. K. Samanta, A comparative study of fuzzy norms on a linear space, Fuzzy##Sets and Systems, 159(6) (2008), 670{684.##[2] T. Bag and S. K. Samanta, Fixed point theorems in Felbin type fuzzy normed linear spaces,##The Journal of Fuzzy Mathematics, to appear.##[3] C. Borelli and G. L. Forti, On a general Hyers{Ulam stability result, Internat. J. Math. Math.##Sci., 18 (1995), 229{236.##[4] S. Czerwik, The stability of the quadratic functional equation, in: Th. M. Rassias, J. Tabor,##eds., Stability of Mappings of Hyers{Ulam Type, Hadronic Press, Florida, (1994), 81{91.##[5] V. A. Faiziev, T. M. Rassias and P. K. Sahoo, The space of ( ; ##) additive mappings on##semigroup, Trans. Amer. Math. Soc., 354(11) (2002), 4455{4472.##[6] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems, 48 (1992),##[7] T. Gantner, R. Steinlage and R. Warren, Compactness in fuzzy topological spaces, J. Math.##Anal. Appl., 62 (1978) 547562.##[8] P. Gavruta, A generalization of the Hyers{Ulam{Rassias stability of approximately additive##mappings, J. Math. Anal. Appl., 184 (1994), 431{436.##[9] U. Hoehle, Fuzzy real numbers as Dedekind cuts with respect to a multiplevalued logic, Fuzzy##Sets and Systems, 24 (1987) 263278.##[10] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A.,##27 (1941), 222{224.##[11] K. Jun, H. Kim and J. M. Rassias, Extended Hyers{Ulam stability for Cauchy{Jensen map##pings, J. Dierence Equ. Appl., 13 (2007), 1139{1153.##[12] K. Jun and Y. Lee, On the Hyers{Ulam{Rassias stability of a Pexiderized quadratic inequal##ity, Math. Inequal. Appl., 4 (2001), 93{118.##[13] S. M. Jung, Hyers{Ulam{Rassias stability of functional equations in nonlinear analysis,##Springer Science, New York, 2011.##[14] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984),##[15] O. Kaleva, The completion of fuzzy metric spaces, J. Math. Anal. Appl., 109 (1985), 194198.##[16] O. Kaleva, A comment on the completion of fuzzy metric spaces, Fuzzy Sets and Systems,##159(16) (2008), 21902192.##[17] P. Kannappan, Functional equations and inequalities with applications, Springer Science,##New York, 2009.##[18] R. Lowen, Fuzzy set theory, Ch. 5 : Fuzzy Real Numbers, Kluwer, Dordrecht, 1996.##[19] A. Maturo, On some structures of fuzzy numbers, Iranian Journal of Fuzzy Systems, 6 (2009),##[20] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy version of HyersUlamRassias theorem,##Fuzzy Sets and Systems, 159(6) (2008), 720729. ##[21] F. Moradlou, H. Vaezi and C. Park, Fixed points and stability of an additive functional##equation of nApollonius type in Calgebras, Abstract and Applied Analysis, 2008, Article##ID 672618, 13 pages, 2008. doi:10.1155/2008/672618.##[22] F. Moradlou, A. Najati and H. Vaezi, Stability of homomorphisms and derivations on C##ternary rings associated to an Euler{Lagrange type additive mapping, Result. Math., 55##(2009), 469486.##[23] M. S. Moslehian, On the orthogonal stability of the Pexiderized quadratic equation, J. Dier##ence Equ. Appl., 11 (2005), 999{1004.##[24] C. Park, Modied Trif 's functional equations in Banach modules over a Calgebra and##approximate algebra homomorphisms, J. Math. Anal. Appl., 278 (2003), 93{108.##[25] C. Park, On an approximate automorphism on a Calgebra, Proc. Amer. Math. Soc., 132##(2004), 1739{1745.##[26] C. Park and T. M. Rassias, Hyers{Ulam stability of a generalized Apollonius type quadratic##mapping, J. Math. Anal. Appl., 322 (2006), 371{381.##[27] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J.##Funct. Anal., 46 (1982), 126{130.##[28] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull.##Sci. Math., 108 (1984), 445{446.##[29] J. M. Rassias, Solution of a problem of Ulam, J. Approx. Theory, 57 (1989), 268{273.##[30] T. M. 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ALGEBRAICALLYTOPOLOGICAL SYSTEMS AND
ATTACHMENTS
ALGEBRAICALLYTOPOLOGICAL SYSTEMS AND
ATTACHMENTS
2
2
The paper continues the study of the authors on relationships between emph{topological systems} of S.~Vickers and emph{attachments} of C.~Guido. We extend topological systems to emph{algebraicallytopological systems}. A particular instance of the latter, called emph{attachment system}, incorporates the notion of attachment, thus, making it categorically redundant in mathematics. We show that attachment systems are equipped with an internal topology, which is similar to the topology induced by locales. In particular, we provide an attachment system analogue of the wellknown categorical equivalence between sober topological spaces and spatial locales.
1
The paper continues the study of the authors on relationships between emph{topological systems} of S.~Vickers and emph{attachments} of C.~Guido. We extend topological systems to emph{algebraicallytopological systems}. A particular instance of the latter, called emph{attachment system}, incorporates the notion of attachment, thus, making it categorically redundant in mathematics. We show that attachment systems are equipped with an internal topology, which is similar to the topology induced by locales. In particular, we provide an attachment system analogue of the wellknown categorical equivalence between sober topological spaces and spatial locales.
65
102
Anna
Frascella
Anna
Frascella
Department of Mathematics E. De Giorgi", University of Salento,
P. O. Box 193, 73100 Lecce, Italy
Department of Mathematics E. De Giorgi",
Italy
frascella anna@libero.it
Cosimo
Guido
Cosimo
Guido
Department of Mathematics E. De Giorgi", University of Salento,
P. O. Box 193, 73100 Lecce, Italy
Department of Mathematics E. De Giorgi",
Italy
cosimo.guido@unisalento.it
Sergey A.
Solovyov
Sergey A.
Solovyov
Department of Mathematics, University of Latvia, Zellu iela 8,
LV1002 Riga, Latvia and Institute of Mathematics and Computer Science, University
of Latvia, Raina bulvaris 29, LV1459 Riga, Latvia
Department of Mathematics, University of
Latvia
solovjovs@fme.vutbr.cz
Algebraicallytopological system
Attachment system
Categoricallyalgebraic topology
Dual attachment pair
Localic algebra
Localification of systems
(Varietybased) pointless topology
Spatialization of systems
Topological theory morphism
Variety
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Preservation theorems in {L}ukasiewicz \model theory
Preservation theorems in {L}ukasiewicz \model theory
2
2
We present some model theoretic results for {L}ukasiewiczpredicate logic by using the methods of continuous model theorydeveloped by Chang and Keisler.We prove compactness theorem with respect to the class of allstructures taking values in the {L}ukasiewicz $texttt{BL}$algebra.We also prove some appropriate preservation theorems concerning universal and inductive theories.Finally, Skolemization and Morleyization in this framework are discussed andsome natural examples of fuzzy theories are presented.
1
We present some model theoretic results for {L}ukasiewiczpredicate logic by using the methods of continuous model theorydeveloped by Chang and Keisler.We prove compactness theorem with respect to the class of allstructures taking values in the {L}ukasiewicz $texttt{BL}$algebra.We also prove some appropriate preservation theorems concerning universal and inductive theories.Finally, Skolemization and Morleyization in this framework are discussed andsome natural examples of fuzzy theories are presented.
103
113
SeyedMohammad
Bagheri
SeyedMohammad
Bagheri
Department of Pure Mathematics, Faculty of Mathemat
ical Sciences, Tarbiat Modares University, P.O. Box 14115134, and Institute for Re
search in Fundamental Sciences (IPM), P. O. Box 193955746, Tehran, Iran
Department of Pure Mathematics, Faculty of
Iran
bagheri@modares.ac.ir
Morteza
Moniri
Morteza
Moniri
Department of Mathematics, Shahid Beheshti University, G. C.,
Evin, Tehran, Iran
Department of Mathematics, Shahid Beheshti
Iran
mmoniri@sbu.ac.ir, ezmoniri@gmail.com
Continuous model theory
{L}ukasiewicz logic
Preservation theorems
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NEW RESULTS ON THE EXISTING FUZZY DISTANCE
MEASURES
NEW RESULTS ON THE EXISTING FUZZY DISTANCE
MEASURES
2
2
In this paper, we investigate the properties of some recently proposed fuzzy distance measures. We find out some shortcomings for these distances and then the obtained results are illustrated by solving several examplesand compared with the other fuzzy distances.
1
In this paper, we investigate the properties of some recently proposed fuzzy distance measures. We find out some shortcomings for these distances and then the obtained results are illustrated by solving several examplesand compared with the other fuzzy distances.
115
124
Saeid
Abbasbandy
Saeid
Abbasbandy
Department of Mathematics, Imam Khomeini International Uni
versity, Ghazvin, 3414916818, Iran
Department of Mathematics, Imam Khomeini
Iran
abbasbandy@yahoo.com
Soheil
Salahshour
Soheil
Salahshour
Young Researchers and Elite Club, Mobarakeh Branch, Islamic
Azad University, Mobarakeh, Iran
Young Researchers and Elite Club, Mobarakeh
Iran
soheilsalahshour@yahoo.com
Fuzzy distance measure
Metric properties
Fuzzy numbers
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representation theorems of $L$subsets and $L$families on complete residuated lattice
representation theorems of $L$subsets and $L$families on complete residuated lattice
2
2
In this paper, our purpose is twofold. Firstly, the tensor andresiduum operations on $L$nested systems are introduced under thecondition of complete residuated lattice. Then we show that$L$nested systems form a complete residuated lattice, which isprecisely the classical isomorphic object of complete residuatedpower set lattice. Thus the new representation theorem of$L$subsets on complete residuated lattice is obtained. Secondly, weintroduce the concepts of $L$family and the system of $L$subsets,then with the tool of the system of $L$subsets, we obtain therepresentation theorem of intersectionpreserving $L$families oncomplete residuated lattice.
1
In this paper, our purpose is twofold. Firstly, the tensor andresiduum operations on $L$nested systems are introduced under thecondition of complete residuated lattice. Then we show that$L$nested systems form a complete residuated lattice, which isprecisely the classical isomorphic object of complete residuatedpower set lattice. Thus the new representation theorem of$L$subsets on complete residuated lattice is obtained. Secondly, weintroduce the concepts of $L$family and the system of $L$subsets,then with the tool of the system of $L$subsets, we obtain therepresentation theorem of intersectionpreserving $L$families oncomplete residuated lattice.
125
136
Hui
Han
Hui
Han
Department of Mathematics, Ocean University of China, 266100 Qingdao,
P.R. China
Department of Mathematics, Ocean University
China
hanhui200801@163.com
Jinming
Fang
Jinming
Fang
Department of Mathematics, Ocean University of China, 266100 Qing
dao, P.R. China
Department of Mathematics, Ocean University
China
jinmingfang@163.com
Complete residuated lattices
$L$subsets
$L$nested systems
$L$families
Level $L$subsets
Representation theorems
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Existence of Extremal Solutions for Impulsive Delay Fuzzy
Integrodifferential Equations in $n$dimensional Fuzzy Vector Space
Existence of Extremal Solutions for Impulsive Delay Fuzzy
Integrodifferential Equations in $n$dimensional Fuzzy Vector Space
2
2
In this paper, we study the existence of extremal solutions forimpulsive delay fuzzy integrodifferential equations in$n$dimensional fuzzy vector space, by using monotone method. Weshow that obtained result is an extension of the result ofRodr'{i}guezL'{o}pez cite{rod2} to impulsive delay fuzzyintegrodifferential equations in $n$dimensional fuzzy vector space.
1
In this paper, we study the existence of extremal solutions forimpulsive delay fuzzy integrodifferential equations in$n$dimensional fuzzy vector space, by using monotone method. Weshow that obtained result is an extension of the result ofRodr'{i}guezL'{o}pez cite{rod2} to impulsive delay fuzzyintegrodifferential equations in $n$dimensional fuzzy vector space.
137
157
Young
Chel Kwun
Young
Chel Kwun
Department of Mathematics, DongA University, Busan 604714,
Republic of Korea
Department of Mathematics, DongA University,
Korea
yckwun@dau.ac.kr
Jeong Soon
Kim
Jeong Soon
Kim
Department of Math. Education, DaeguUniversity, Gyeongsan 712
714, Republic of Korea
Department of Math. Education, DaeguUniversity,
Korea
jeskim@donga.ac.kr
Jin Han
Park
Jin Han
Park
Department of Applied Mathematics, Pukyong National University,
Buan 608737, Republic of Korea
Department of Applied Mathematics, Pukyong
Korea
jihpark@pknu.ac.kr
Extremal solution
Impulsive delay fuzzy integrodifferential equation
$n$dimensional fuzzy vector space
Monotone method
[[1] P. Balasubramaniam and S. Muralisankar, Existence and uniqueness of fuzzy solution for##semilinear fuzzy integrodierential equations with nonlocal conditions, Computer & Mathe##matics with Applications, 47 (2004), 1115{1122.##[2] P. Diamond and P. Kloeden, Metric spaces of fuzzy sets, World Scientic, 1994.##[3] Y. C. Kwun, J. S. Kim, M. J. Park and J. H. Park, Nonlocal controllability for the semilinear##fuzzy integrodierential equations in ndimensional fuzzy vector space, Advances in Dierence##Equations, Article ID734090, 2009 (2009).##[4] Y. C. Kwun, J. S. Kim, M. J. Park and J. H. Park, Controllability for the impulsive semilinear##nonlocal fuzzy integrodierential equations in ndimensional fuzzy vector space , Advances##in Dierence Equations, Article ID983483, 2010(2010).##[5] Y. C. Kwun, J. S. Kim and J. H. Park, Existence of extremal solutions for impulsive fuzzy dif##ferential equations with periodic boundary value in ndimensional fuzzy vector space, Journal##of Computational Analysis and Applications, 13 (2011), 1157{1170.##[6] J. J. Nieto and R. RodrguezLopez, Existence of extrmal solutions for quadratic fuzzy equa##tions, Fixed Point Theory Appl., 3 (2005), 321{342.##[7] R. RodrguezLopez, Periodic boundary value problems for impulsive fuzzy dierential equa##tions Fuzzy Sets and Systems, 159 (2008), 1384{1409.##[8] R. RodrguezLopez, Monotone method for fuzzy dierential equations Fuzzy Sets and Sys##tems, 159 (2008), 2047{2076.##[9] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (1987), 319{330##[10] O. Solaymani Fard and A. Vahidian Kamyad, Modied kstep method for solving fuzzy initial##value problems, Iranian Journal of Fuzzy Systems, 8 (2011), 49{63.##[11] G. Wang, Y. Li and C. Wen, On fuzzy ncell number and ndimension fuzzy vectors, Fuzzy##Sets and Systems, 158 (2007), 71{84.##[12] G. Wang and C. Wu, Fuzzy ncell number and the dierential of fuzzy ncell number value##mappings, Fuzzy Sets and Systems, 130 (2002), 367{381.##]
On fuzzy convex latticeordered subgroups
On fuzzy convex latticeordered subgroups
2
2
In this paper, the concept of fuzzy convex subgroup (resp. fuzzy convex latticeordered subgroup) of an ordered group (resp. latticeordered group) is introduced and some properties, characterizations and related results are given. Also, the fuzzy convex subgroup (resp. fuzzy convex latticeordered subgroup) generated by a fuzzy subgroup (resp. fuzzy subsemigroup) is characterized. Furthermore, the Fundamental Homomorphism Theorem is established. Finally, it is proved that the class of all fuzzy convex latticeordered subgroups of a latticeordered group $G$ forms a complete Heyting sublattice of the lattice of fuzzy subgroups of $G$.
1
In this paper, the concept of fuzzy convex subgroup (resp. fuzzy convex latticeordered subgroup) of an ordered group (resp. latticeordered group) is introduced and some properties, characterizations and related results are given. Also, the fuzzy convex subgroup (resp. fuzzy convex latticeordered subgroup) generated by a fuzzy subgroup (resp. fuzzy subsemigroup) is characterized. Furthermore, the Fundamental Homomorphism Theorem is established. Finally, it is proved that the class of all fuzzy convex latticeordered subgroups of a latticeordered group $G$ forms a complete Heyting sublattice of the lattice of fuzzy subgroups of $G$.
159
172
Mahmood
Bakhshi
Mahmood
Bakhshi
Department of Mathematics, Bojnord University, Bojnord, Iran
Department of Mathematics, Bojnord University,
Iran
bakhshi@ub.ac.ir
Latticeordered group
Convex subgroup
Fuzzy convex subgroup
[[1] N. Ajmal and K. V. Thomas, Fuzzy lattices, Information Sciences, 79 (1994), 271{291.##[2] J. M. Anthony and H. Sherwood, Fuzzy groups redened, J. Math. Anal. Appl., 69 (1979),##[3] J. M. Anthony and H. Sherwood, A characterization of fuzzy subgroups, Fuzzy Sets and##Systems, 7 (1982), 297{305.##[4] T. S. Blyth, Lattices and ordered algebraic structures, SpringerVerlag, London, 2005.##[5] Y. Bo and W. Wangming, Fuzzy ideals on a distributive lattice, Fuzzy Sets and Systems, 35##(1990), 231{240.##[6] P. S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl., 84 (1981), 264{269.##[7] D. S. Malik, J. N. Mordeson and P. S. Nair, Fuzzy generators and fuzzy direct sums of abelian##groups, Fuzzy Sets and Systems, 50 (1992), 193{199.##[8] J. N. Mordeson, K. R. Bhutani and A. Rosenfeld, Fuzzy group theory, SpringerVerlag,##Netherlands, 2005.##[9] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512{517.##[10] G. S. V. Satya Saibaba, Fuzzy lattice ordered groups, Southeast Asian Bull. Math., 32 (2008),##[11] U. M. Swamy and D. Viswanadha Raju, Fuzzy ideals and congruences of lattices, Fuzzy Sets##and Systems, 95 (1998), 249{253.##[12] Y. Yin, Y. B. Jun and Z. Yang, More general forms (; )fuzzy ideals of ordered semigroups,##Iranian Journal of Fuzzy Systems, 9(4) (2012), 99113.##[13] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{353.##]
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