2015
4
3
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An algorithm for determining common weights by concept of membership function
2
2
Data envelopment analysis (DEA) is a method to evaluate the relative efficiency of decision making units (DMUs). In this method, the issue has always been to determine a set of weights for each DMU which often caused many problems. Since the DEA models also have the multiobjective linear programming (MOLP) problems nature, a rational relationship can be established between MOLP and DEA problems to overcome the problem of determining weights. In this study, a membership function was defined base on the results of CCR model and cross efficiency, and by using this membership function in a proposed model, we obtained a common set of weights for all DMUs. Finally, by solving a sample problem, the proposed algorithm was explained.
1

165
172


S.
Saati
Department of Mathematics, North Tehran Branch, Islamic Azad University, Tehran, Iran
Department of Mathematics, North Tehran Branch,
Iran
s_saatim@iautnb.ac.ir


N.
Nayebi
Department of Mathematics, North Tehran Branch,
Islamic Azad University, Tehran, Iran
Department of Mathematics, North Tehran Branch,
Is
Iran
Data Envelopment Analysis (DEA)
Cross efficiency
Membership function
Common Set of Weights
Multiobjective programming problem
[[1] A. Charnes, W. W. Cooper, Management models and industrial applications of linear programming, John Wiley, New York, 1961. ##[2] A. Charnes, W. W. Cooper and E. Rhodes, Measuring the efficiency of decision making units, European journal of operational research. (1978), 2, 429444. ##[3] G. R. Jahanshahloo, A. Memariani, F. Hosseinzadeh Lotfi and H. Z. Rezai, A note on some of DEA models and finding efficiency and complete ranking using common set of weights, Applied mathematics and computation. (2005), 166, 265281. ##[4] J. Lu, G. Zhang, D. Ruan and F. Wu, Multiobjective group decision making methods software and applications with fuzzy set techniques. Imperial collage press. 2007. ##[5] Y. Roll, W. Cook and B. Golany, Controlling factor weights in data envelopment analysis. IIE Trans, (1991), 23, 29. ##[6] T. Roll, B. Golany, Alternate methods of treating factor weights in data envelopment analysis, Omega, (1993), 21, 99109. ##[7] M. S. Saati, Determining a common set of weights in DEA by solving a linear programming, Journal of industrial engineering international, (2008), 6, 5156. ##[8] T. R. Sexton, R. H. Silkman and A. J. Hogan, Data envelopment analysis: critique and extensions. In: R.H. Silkman (Ed.), Measuring efficiency: An assessment of data envelopment analysis. San Francisco, CA, JosseyBass, 1986. ##[9] Y. M. Wang, K. P. Chin, Some alternative models for DEA cross efficiency evaluation, Production economics, (2010), 128, 33233. ##[10] Y. M. Wand, K. S. Chin and Y. Luo, Cross efficiency evaluation based on ideal and antiideal decision making units, Expert systems with applications, (2011), 38, 1031210319.##]
Application of triangular functions for solving the vasicek model
2
2
This paper introduces a numerical method for solving the vasicek model by using a stochastic operational matrix based on the triangular functions (TFs) in combination with the collocation method. The method is stated by using conversion the vasicek model to a stochastic nonlinear system of $2m+2$ equations and $2m+2$ unknowns. Finally, the error analysis and some numerical examples are provided to demonstrate applicability and accuracy of this method.
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173
182


Z.
Sadati
Department of Mathematics, Khomein Branch, Islamic
Azad University, Khomein, Iran
Department of Mathematics, Khomein Branch,
Iran
sadatizahra501@gmail.com


Kh.
Maleknejad
Department of Mathematics, Khomein Branch, Islamic
Azad University, Khomein, Iran
Department of Mathematics, Khomein Branch,
Iran
zahra_sadati47@yahoo.com
Triangular functions
Stochastic operational matrix
Vasicek model
collocation method
[[1] A. Deb, A. Dasgupta and G. Sarkar, A new set of orthogonal functions and its application to the analysis of dynamic systems. J. Franklin Inst, vol. 343, (2006) 126. ##[2] B. Oksendal, Stochastic Differential Equations, An Introduction with Applications, Fifth Edition, SpringerVerlag, New York, 1998. ##[3] E. Babolian, H. R. Marzban, and M. Salmani, Using triangular orthogonal functions for solving fredholm integral equations of the second kind, Appl. Math. Comput, vol. 201, (2008) 452456. ##[4] E. Babolian, Z. Masouri and S. HatamzadehVarmazya, A direct method for numerically solving integral equations system using orthogonal triangular functions, Int. J. Industrial Mathematics, vol. 1, no. 2, (2009) 135145, 2009. ##[5] E. Pardoux and P. Protter, Stochastic volterra equations with anticipating coefficients, Ann. Probab, vol. 18, (1990) 16351655. ##[6] K. Maleknejad, H. Almasieh and M. Roodaki, Triangular functions (TF) method for the solution of volterrafredholm integral equations, Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 11, (2009) 32933298. ##[7] K. Maleknejad, and Z. Jafari Behbahani, Applications of twodimensional triangular functions for solving nonlinear class of mixed volterrafredholm integral equations, Mathematical and Computer Modelling. ##[8] K. Maleknejad, M. Khodabin, and M. Rostami, A numerical method for solving mdimensional stochastic Itvolterra integral equations by stochastic operational matrix, Computers and Mathematics with Applications, vol. 63, (2012) 133143. ##[9] K. Maleknejad, M. Khodabin and M. Rostami, Numerical solution of stochastic volterra integral equations by stochastic operational matrix based on block pulse functionsx, Mathematical and Computer Modelling, vol. 55, (2011) 791800. ##[10] M. A. Berger and V.J. Mizel, Volterra equations with Ito integrals I, J. Integral Equations vol. 2, no. 3, (1980) 187245. ##[11] M. Khodabin, K. Maleknejad and F. Hosseini, Application of triangular functions to numerical solution of stochastic volterra integral equations, IAENG International Journal of Applied Mathematics, 2013, IJAM 43101. ##[12] P. E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, Applications of Mathematics, SpringerVerlag, Berlin, 1999.##]
Lie higher derivations on $B(X)$
2
2
Let $X$ be a Banach space of $dim X > 2$ and $B(X)$ be the space of bounded linear operators on X. If $L : B(X)to B(X)$ be a Lie higher derivation on $B(X)$, then there exists an additive higher derivation $D$ and a linear map $tau : B(X)to FI$ vanishing at commutators $[A, B]$ for all $A, Bin B(X)$ such that $L = D + tau$.
1

183
192


S.
Ebrahimi
Department of Mathematics, Payame Noor University, P.O. Box 193953697, Tehran, Iran.
Department of Mathematics, Payame Noor University,
Iran
seebrahimi2272@gmail.com
Lie derivation
Lie higher derivations
higher derivations
[[1] W. Cheung,Lie derivations on triangular matrices, Linear Multilinear Algebra 55 (2007), 619.626. ##[2] Y. Du, Y. Wang,Lie derivations of generalized matrix algebras, Linear Algebra Appl. 433 (2012), 27192726. ##[3] M. Ferrero, C. Haetinger,Higher derivations of semiprime rings, Comm. Algebra. 30 (2002), 23212333. ##[4] F. Lu ., B. Liu,Lie derivable maps on B(X) , J. Math. Anals. Appl. 372(2010), 369376. ##[5] Y. B. Li, Z. K. Xiao, Additivity of maps on generalized matrix algebras, Electron. J. Linear Algebra 22, (2011) 743757. ##[6] A. Nakajima, it On generalized higher derivations, Turk. J. Math. 24(2000), 295311. ##[7] X. F. Qi,chacterization of Lie higher derivation on triangular algebras, Acta Mathematica Sinica. 26 (2013), 10071018. ##[8] G. A. Swain, P.S. Blau,Lie derivations in prime rings with involution , Canad. Math. Bull. 42, (1999) 401411. ##[9] Z.K Xiao, F. Wei,Jordan higher derivations on triangular algebras, Linear Algebra and its Applications. 432 (2010), 26152622. ##[10] Q. X. Fei, H. J. Chen, Lie higher derivations on nest algebras, Comm. Math. Research.(2010) 131143. ##[11] Z.K Xiao, F. Wei , Higher derivations of triangular algebras and its generalizations, Linear Algebra Appl. 435 (2011),10341054. ##[12] W. Y. Yu, J. H. Zhang, Chacterization of Lie higher and Lie triple derivation on triangular algebras, J. Korean Math. Soc. 49 (2012), 419433.##]
Bernoulli collocation method with residual correction for solving integralalgebraic equations
2
2
The principal aim of this paper is to serve the numerical solution of an integralalgebraic equation (IAE) by using the Bernoulli polynomials and the residual correction method. After implementation of our scheme, the main problem would be transformed into a system of algebraic equations such that its solutions are the unknown Bernoulli coefficients. This method gives an analytic solution when the exact solutions are polynomials. Also, an error analysis based on the use of the Bernoulli polynomials is provided under several mild conditions. Several examples are included to illustrate the efficiency and accuracy of the proposed technique and also the results are compared with the different methods.
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193
208


F.
Mirzaee
Faculty of Mathematical Sciences and Statistics, Malayer University,
P. O. Box 6571995863, Malayer, Iran
Faculty of Mathematical Sciences and Statistics,
Iran
f.mirzaee@malayeru.ac.ir
Integral algebraic equations
Approximate solutions
Bernoulli collocation method
error analysis
[[1] V. Balakumar., K. Murugesan, Numerical solution of Volterra integralalgebraic equations using block pulse functions, Appl. Math. Comput., 263 (2015), pp. 165170. ##[2] J. Biazar, M. Eslami, Modified HPM for solving systems of Volterra integral equations of the second kind, J. King Saud Univ. Sci., 23 (1) (2011), pp. 3539. ##[3] H. Brunner, Collocation methods for Volterra integral and related functional equations, University Press, Cambridge, 2004. ##[4] O. S. Budnikova, M. V. Bulatov, Numerical solution of integralalgebraic equations for multistep methods, Comput. Math. Math. Phys., 52(5) (2012), pp. 691701. ##[5] M. V. Bulatov, V. F. Chistyakov, The properties of differentialalgebraic systems and their integral analogs, Memorial Uni. Newfoundland, preprint, 1997. ##[6] J. R. Cannon, The Onedimensional heat equation, Cambridge Uni. Press, New York, 1984. ##[7] C. W. Gear, Differentialalgebraic equations, indices, and integralalgebraic equations, SIAM. J. Numer. Anal., 27 (1990), pp. 15271534. ##[8] C. W. Gear, Differentialalgebraic equations and index transformation, SIAM. J. Stat. Comput., 1 (1988), pp. 3947. ##[9] H. G. Golub, C.F. Van Loan, Matrix computations, Johns Hopkins Uni. Press, London, 1996. ##[10] A. M. Gomilko, A Dirichlet problem for the biharmonic equation in a semiinfinite strip, J. Eng. Math., 46 (2003), pp. 253268. ##[11] J. Janno, L. von Wolfersdorf, Inverse problems for identification of memory kernels in viscoelasticity, Math. Meth. Appl. Sci., 20 (1997), pp. 291314. ##[12] B. Jumarhon, W. Lamb, S. McKee, T. Tang, A Volterra integral type method for solving a class of nonlinear initialboundary value problems, Numer. Meth. Partial Diff. Eq., 12 (1996), pp. 265281. ##[13] V. V. Kafarov, B. Mayorga, C. Dallos, Mathematical method for analysis of dynamic processes in chemical reactors, Chem. Eng. Sci., 54 (1999), pp. 46694678. ##[14] J. P. Kauthen, The numerical solution of integralalgebraic equations of index1 by polynomial spline collocation methods, Math. Comp., 236 (2000), pp. 15031514. ##[15] F. Mirzaee, Numerical computational solution of the linear Volterra integral equations system via rationalized Haar functions, J. King Saud Uni. Sci., 22 (4) (2010), pp. 265268. ##[16] F. Mirzaee, S. F. Hoseini, Solving systems of linear Fredholm integrodifferential equations with Fibonacci polynomials, Ain Shams Engin. J., 5 (2014), pp. 271283. ##[17] F. Mirzaee, S. F. Hoseini, A Fibonacci collocation method for solving a class of FredholmVolterra integral equations in twodimensional spaces, BeniSuef Uni. J. Bas. App. Sci., 3 (2014), pp. 157163. ##[18] P. Natalini, A. Bernaridini, A Generalization of the Bernoulli polynomials, J. Appl. Math., 3 (2003), pp. 155163. ##[19] M. Rabbani, K. Maleknejad, N. Aghazadeh, Numerical computational solution of the Volterra integral equations system of the second kind by using an expansion method, Appl. Math. Comput., 187 (2) (2007), pp. 11431146. ##[20] G. Rzadkowski, S. Lepkowski, A Generalization of the EulerMaclaurin Summation Formula: An Application to Numerical Computation of the FermiDirac Integrals, J. Sci. Comput., 35 (2008), pp. 6374. ##[21] S. Pishbin, F. Ghoreishi, M. Hadizadeh, A posteriori error estimation for the Legendre collocation method applied to integralalgebraic Volterra equations, Electron. Trans. Numer. Anal., 38 (2011), pp. 327346. ##[22] N. Sah´yn,S.Y¨uzbas´y,M.G¨ulsu, A collocation approach for solving systems of linear Volterra integral equations with variable coefficients, Comput. Math. Appl., 62(2) (2011), pp. 755769. ##[23] Stewart, G.W. Matrix algorithms, Volume I: Basic Decompositions, SIAM, Philadelphia, 1998. ##[24] A. Tahmasbi, O.S. Fard, Numerical solution of linear Volterra integral equations system of the second kind, Appl. Math. Comput., 201 (1) (2008), pp. 547552. ##[25] E. Tohidi, A. H. Bhrawy, K. Erfani, A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation, Appl. Math. Mod., 37(6) (2012), pp. 42834294. ##[26] F. Toutounian, E. Tohidi, S. Shateyi, A collocation method based on the Bernoulli operational matrix for solving highorder linear complex differential equations in a rectangular domain, Abst. Appl. Anal., Hindawi Pub. Co., 2013 (2013), pp. 112. ##[27] L. V. Wolfersdorf, On identification of memory kernel in linear theory of heat conduction, Math. Meth. Appl. Sci., 17 (1994), pp. 919932. ##[28] L. H. Yang, J. H. Shen, Y. Wang, The reproducing kernel method for solving system of the linear Volterr aintegral equations with variable coefficients, J. Comput. Appl. Math., 236 (2012), pp. 23982405. ##[29] A. I. Zenchuk, Combination of inverse spectral transform method and method of characteristics: Deformed Pohlmeyer equation, J. Nonlinear Math. Phys., 15 (2008), pp. 437448.##]
On the girth of the annihilatingideal graph of a commutative ring
2
2
The annihilatingideal graph of a commutative ring $R$ is denoted by $AG(R)$, whose vertices are all nonzero ideals of $R$ with nonzero annihilators and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=0$. In this article, we completely characterize rings $R$ when $gr(AG(R))neq 3$.
1

209
216


M.
Ahrari
Department of Mathematics, Islamic Azad University,
Central Tehran Branch, Tehran, Iran
Department of Mathematics, Islamic Azad University
Iran
mar.ahrari.sci@iauctb.ac.ir


Sh. A.
Safari Sabet
Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran
Department of Mathematics, Islamic Azad University
Iran
sh_safarisabet@iauctb.ac.ir


B.
Amini
Department of Mathematics, College of Sciences, Shiraz University, Shiraz, Iran
Department of Mathematics, College of Sciences,
Iran
bamini@shirazu.ac.ir
annihilatingideal graph
star graph
bipartite graph
girth
[[1] G. Aalipour, S. Akbari, M. Behboodi, R. Nikandish, M. J. Nikmehr and F. Shaveisi, The classification of annihilatingideal graphs of commutative rings, Algebra Colloquium 21(2) (2014) 249256. ##[2] A. Amini, B. Amini, E. Momtahan and M. H. Shirdareh Haghighi, On a graph of ideals, Acta Math. Hungar 134 (3) (2012) 369–384. ##[3] D. F. Anderson, M. C. Axtell and J. A. Stickles, Zerodivisor graphs in commutative rings, in Commutative Algebra, Noetherian and NonNoetherian Perspective, eds. M. Fontana, S.E. Kabbaj, B. Olberding and I. Swanson (SpringVerlag, New York, 2011), 2345. ##[4] D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra 320(7) (2008) 27062719. ##[5] N. Ashrafi, H. R. Maimani, M. R. Pouranki and S. Yassemi, Unit graphs associated with rings, Comm. Algebra 38 (2010) 28512871. ##[6] M. Baziar, E. Momtahan and S. Safaeeyan, A zerodivisor graph for modules with respect to their (first) dual, J. Algebra Appl. 12(2) (2013) 1250151. ##[7] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988) 208–226. ##[8] M. Behboodi and Z. Rakeei, The annihilatingideal graph of commutative rings I, J. Algebra Appl. 10(4) (2011) 727739. ##[9] M. Behboodi and Z. Rakeei, The annihilatingideal graph of commutative rings II, J. Algebra Appl. 10(4) (2011) 741753. ##[10] S. P. Redmond, The zerodivisor graph of a noncommutative ring, Int. J. Commut. Rings, 1(4) (2002) 203211.##]
AnalyticalApproximate Solution for Nonlinear Volterra IntegroDifferential Equations
2
2
In this work, we conduct a comparative study among the combine Laplace transform and modied Adomian decomposition method (LMADM) and two traditional methods for an analytic and approximate treatment of special type of nonlinear Volterra integrodifferential equations of the second kind. The nonlinear part of integrodifferential is approximated by Adomian polynomials, and the equation is reduced to a simple equations. The proper implementation of combine Laplace transform and modified Adomian decomposition method can extremely minimize the size of work if compared to existing traditional techniques. Moreover, three particular examples are discussed to show the reliability and the performance of method.
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217
228


M.
Matinfar
Department of Mathematics, University of Mazandaran, Babolsar, PO. Code 4741695447, Iran
Department of Mathematics, University of
Iran
m.matinfar@umz.ac.ir


A.
Riahifar
Department of Mathematics, University of Mazandaran, Babolsar, PO. Code 4741695447, Iran
Department of Mathematics, University of
Iran
abbas.riahifar@yahoo.com
Nonlinear Volterra integrodifferential equations
Laplace Transform method
Modified Adomian decomposition method
[[1] A. M. Wazwaz, A comparison study between the modified decomposition method and the traditional methods for solving nonlinear integral equations, Appl. Math. Comput. 181 (2006),17031712. ##[2] A. M. Wazwaz, A First Course in Integral Equations, World Scientific, 1997. ##[3] F. G. Tricomi, Integral Equations, Dover, 1982. ##[4] G. Adomian, A review of the decomposition method and some recent results for nonlinear equation, Math. Comput. Model. 13 (1992), 1743. ##[5] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, MA, 1994. ##[6] H. R. Thieme, A model for the spatio spread of an epidemic, J. Math. Biol. 4 (1977), 337351. ##[7] J. Manafianheris, Solving the integrodifferential equations using the odified Laplace Adomian Decomposition Method, Journal of Mathematical Extension. 6 (2012), 6579. ##[8] M. Hussain, M. Khan, Modified Laplace decomposition method, Appl. Math. Scie. 4 (2010),17691783. ##[9] M. Tatari, M. Dehghan, Numerical solution of Laplace equation in a disk using the Adomian decomposition method, Phys. Scr. 72 (2005), 345348. ##[10] N. Ngarhasta, B. Some, K. Abbaoui, and Y. Cherruault, New numerical study of adomian method applied to a diffusion model, Kybernetes. 31 (2002), 6175. ##[11] S. A. Khuri, A Laplace decomposition algorithm applied to class of nonlinear differential equations, J. Math. Appl. 4 (2001), 141155. ##[12] S. Hyder A. M. Shah, A. W. Shaikh, and S. H. Sandilo, Modified decomposition method for nonlinear VolterraFredholm integrodifferential equation, Journal of Basic and Applied Sciences. 6 (2010), 1316.##]
On fuzzy soft connected topological spaces
2
2
In this work, we introduce notion of connectedness on fuzzy soft topological spaces and present fundamentals properties. We also investigate effect to fuzzy soft connectedness. Moreover, $C_i$connectedness which plays an important role in fuzzy topological space extend to fuzzy soft topological spaces.
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229
240


S.
Karataş
Department of Mathematics, Ordu University, 52200, Turkey
Department of Mathematics, Ordu University,
Turkey
posbiyikliadam@gmail.com


B.
Kılıccedil
Department of Mathematics, Ordu University, 52200, Turkey
Department of Mathematics, Ordu University,
Turkey
burakkilic61@hotmail.com


M.
Tellioğlu
Department of Mathematics, Ordu University, 52200, Turkey
Department of Mathematics, Ordu University,
Turkey
m.uykun.tellioglu@outlook.com
Fuzzy soft set
fuzzy soft topological space
fuzzy soft connectedness
[[1] B. Ahmad, and A. Kharal, On fuzzy soft sets, Advances in Fuzzy Systems, 1 (2009), 16. ##[2] N. Ajmal and J.K. Kohli, Connectedness in fuzzy topological spaces, Fuzzy Sets and Systems, 31 (1989), 369388. ##[3] H. Akta¸s and N. Cagman, Soft sets and soft group, Information Science, 177 (2007), 27262735. ##[4] A. Aygunoglu and H. Aygun, Some notes on soft topological spaces, Neural Comput and Appl., 21(1) (2011), 113119. ##[5] N. Cagman, and S. Enginoglu, Soft set theory and uniint decision making, European Journal of Operational Research, 207 (2010), 848855. ##[6] N. Cagman, F. Cıtak and S. Enginoglu, Fuzzy parameterized fuzzy soft set theory and its applications, Turk. J. Fuzzy Syst., 1(1) (2010), 2135. ##[7] N. Cagman, S. Karatas and S. Enginoglu, Soft topology, Computers and Mathematics with Applications, 62 (2011), 351358. ##[8] N. Cagman, S. Enginoglu and F. Cıtak, Fuzzy soft set theory and its applications, Iranian Journal of Fuzzy Systems, 8(3) (2011), 137147. ##[9] N. Cagman, Contributions to the theory of soft sets, Journal of New Results in Science, 4 (2014), 3341. ##[10] C. L. Chang, Fuzzy topological spaces, J. Math. Appl., 24 (1968), 182193. ##[11] P.K. Gain, P. Mukherjee, R.P. Chakraborty and M. Pal, On some structural properties of fuzzy soft topological spaces, Intern. J. Fuzzy Mathematical Archive, 1 (2013), 115. ##[12] C¸. Gunduz and S. Bayramov, Some results on fuzzy soft topological spaces Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 835308. ##[13] S. Hussain and B. Ahmad, Some properties of soft topological spaces, Computers and Mathematics with Applications, 62 (2011), 40584067. ##[14] S. Hussain, A note on soft connectedness, Journal of Egyptian Mathematical Society, 23(1) 2015, 611. ##[15] A. Kharal and B. Ahmad, Mappings on fuzzy soft classes, Hindawi Publishing Corporation, Advances in Fuzzy Systems, Article ID 407890, 2009. ##[16] P. K. Maji, R. Biswas and A. R. Roy, Fuzzy soft sets, J. Fuzzy Math., 9(3) (2001), 589602. ##[17] D. Molodtsov, Soft set theoryfirst results, Computers Math. Appl., 37(5) (1999), 1931. ##[18] E. Peyghan, B. Samadi and A. Tayebi, About soft topological spaces Journal of New Results in Science, 2 (2013), 6075. ##[19] S. Roy and T.K. Samanta, A note on fuzzy soft topological spaces, Annals of Fuzzy Mathematics and Informatics, 3(2) (2012), 305311. ##[20] M. Shabir and M. Naz, On soft topological spaces, Computers and Mathematics with Applications, 61 (2011), 17861799. ##[21] T. Simsekler and S¸. Yuksel, Fuzzy soft topological spaces, Annals of Fuzzy Mathematics and Informatics, 5(1) (2013), 8796. ##[22] B. Tanay and M. B. Kandemir, Topological structures of fuzzy soft sets, Computers and Mathematics with Applications, 61 (2011) 412418. ##[23] B. P. Varol and H. Ayg¨un, Fuzzy soft topology, Hacettepe Journal of Mathematics and Statistics Volume 41 (3)(2012), 407419. ##[24] L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338353. ##[25] X. D. Zhao, Connectedness on fuzzy topological spaces, Fuzzy Sets and Systems, 20 (1986), 223240. ##[26] I. Zorlutuna, M. Akdag, W. K. Min, S. Atmaca, Remarks on soft topological spaces, Annals of Fuzzy Mathematics and Informatics, 3(2) (2012), 171185. Rings, 1(4) (2002) 203211.##]