#### Volume-7 ~ Issue-1

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Paper Type |
: | Research Paper |

Title |
: | Case Study for Bank ATM Queuing Model |

Country |
: | Bangladesh |

Authors |
: | S. K. Dhar, Tanzina Rahman |

: | 10.9790/5728-0710105 |

**Abstract:** Queuing is the common activity of customers or people to avail the desired service, which could be processed or distributed one at a time. Bank ATMs would avoid losing their customers due to a long wait on the line. The bank initially provides one ATM in every branch. But, one ATM would not serve a purpose when customers withdraw to use ATM and try to use other bank ATM. Thus the service time needs to be improved to maintain the customers. This paper shows that the queuing theory used to solve this problem. We obtain the data from a bank ATM in a city. We then derive the arrival rate, service rate, utilization rate, waiting time in the queue and the average number of customers in the queue based on the data using Little's theorem and M/M/I queuing model. The arrival rate at a bank ATM on Sunday during banking time is 1 customer per minute (cpm) while the service rate is 1.50 cpm. The average number of customer in the ATM is 2 and the utilization period is 0.70. We conclude the paper by discussing the benefits of performing queuing analysis to a busy ATM.

**Keywords: **Bank ATM, Little's theorem, M/M/I queuing model, Queue, Waiting lines.

[1] N. K. Tiwari & Shishir K. Shandilya, Oprations Research. Third Edition, ISBN 978-81-203-2966-9. Eastern Economy Edition, 2009.

[2] Nita H. Shah, Ravi M. Gor, Hardik Soni, Operations Research. 4th Edition, ISBN 978-81- 203-3128-0. Eastern Economy Edition, 2010.

[3] K. Sanjay, Bose, "An Introduction to queuing system", Springer, 2002.

[4] J. D. C. Little, "A Proof for the Queuing Formula: ", Operations Research, vol. 9(3), 1961, pp. 383-387, doi: 10.2307/167570

[5] M. Laguna and J. Marklund, Business Process Modeling, Simulation and Dsign. ISBN 0-13-091519-X. Pearson Prentice Hall, 2005.

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Paper Type |
: | Research Paper |

Title |
: | Jordan Higher K-Centralizer on -Rings |

Country |
: | Iraq |

Authors |
: | Salah M. Salih, Ali M.Kamal, Balsam M. Hamad |

: | 10.9790/5728-0710614 |

**Abstract:** Let M be a semiprime -ring satisfying a certain assumption. Then we prove that every Jordan left
higher k-centralizer on M is a left higher k-centralizer on M. We also prove that every Jordan higher kcentralizer
of a 2-torsion free semiprime -ring M satisfying a certain assumption is a higher k-centralizer.

**Keywords: **Semiprime -ring, left higher centralizer, higher k-centralizer, Jordan higher k-centralizer𝜏)-centralizr

[1]. R.Awtar, Lie Ideals and Jordan Derivation of Prime Rings, Proc.Amer.Math.Soc., Vol.90, No.1, pp.9-14, 1984.

[2]. W.E.Barnes, On the -Rings of Nobusawa, Pacific J.Math., 18 (1966), 411-422.

[3]. F.J.Jing, On Derivations of -Rings, QuFuShiFan Daxue Xuebeo Ziran Kexue Ban, Vol.13, No.4, pp.159-161, 1987.

[4]. B.E.Johnson and A.M.Sinclair, Continuity of Derivation and a Problem of Kaplasky, Amer.J.Math., 90(1968), 1067-1078.

[5]. M.F.Hoque and A.C.Paul, On Centralizers of Semiprime Gamma Rings, International Mathematical Forum, 6(13)(2011), 627-

638.

[6]. A.H.Majeed and S.M.Salih, Jordan Higher Derivation on Prime -Rings, College of Education, Conference, 16th, Al-

Mustansirhyah Univ., 2009.

[7]. N.Nobusawa, On a Generalization of the Ring Theory, Osaka J. Math., (1964), 81-89.

[8]. M.Sapanci and A.Nakajima, Jordan Derivations on Completely Prime Gamma Rings, Math. Japonica, 46(1)(1997), 47-51.

[9]. S.M.Salih, B.Majid, (U,M) Derivation -Rings, Education College Conference, Al-Mustansiriya University, 2012.

[10]. Z.Ullah and M.Anwar Chaudhary, on K-Centralizer of Semiprime Gamma Rings, Bahauddin Zakariya Univ.Multan, Pakistan,

Vol.6, No.21, 2012.

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Paper Type |
: | Research Paper |

Title |
: | L Inequalities Concerning Polynomials Having Zeros in Closed Interior of A Circle |

Country |
: | India |

Authors |
: | K. K. Dewan, C. M. Upadhye |

: | 10.9790/5728-0711519 |

**Abstract:** Let
0
( )
n
j
i
j
p z a z
be a polynomial of degree n and p(z) be its derivative, then Zygmund [9]
proved that
1 1
2
0
2
'( ) | ( ) | , 1
0
| |
r r i r i r p e d n p e d r
In this paper we shall obtain similar type of inequalities in reverse order for the polynomials having r fold
zeros at origin and rest of the zeros in | z | k , k 1.
Mathematics Subject Classification (2010): 30A10, 30C15, 30C10

**Key words:** Polynomials, Zeros, Polar derivative, Inequality

[1] A.Aziz and W.M.Shah, Inequalit ies for a polynomial and its derivat ives, Math. Ineq. Appl. 7(2004), 379-391.

[2] S. Bernstein, Lecons Sur Les Proprietes extremales et la meilleure approximation des fonctions analytiques d reele, Paris, 1926. W.

[3] T.C. Chan and M.A. Malik, on Erdos-Lax theorem, Proc. Indian Acad. Sci. 92(3) (1983), 191-193.

[4] K.K. Dewan et al., Some inequalit ies for the polar derivat ive of a polynomial, Southeast Asian Bull. Math. 34(2010), 69-77.

[5] V.K. Jain, Integral inequalit ies for polynomials having a zero of order m at the origin, Glasnik Matematicki 37(57) (2002), 83-88.

[6] N.A. Rather, Extremal Properties and Location of the Zeros of Polynomials, Ph.D. Thesis, University of Kashmir, 1998.

[7] Rudin, Real and Complex Analysis, Tata McGraw-Hill Publishing Company (reprinted in India), 1977.

[8] A.E. Taylor, Introduction to Functional Analysis, John Wiley and Sons, Inc., New York, 1958.

[9] Zygmund, A remark on conjugate series, Proc. London Math. Soc. 34 (1932), 392-400.

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**Abstract:** The effects of variable Fluid Properties like variation of permeability, porosity, thermal conductivity and magnetic field on Mixed Convection Heat transfer from Vertical Heated Plate Embedded in a Sparsely Packed Porous Medium have been approached numerically. The boundary layer flow in the porous medium is governed by Lapwood – Forchheimer – Brinkman extended Darcy model and the Lorentz force. The natures of these equations are highly non-linear and coupled each other. The non-linear differential equations are non-dimensionalised using the non-dimensional parameter involving Grashoff number Gr, Prandtl number Pr, Hartmann number M, Eckert number E and so on. Similarity transformations are employed and the resulting ordinary differential equations are solved numerically by using shooting algorithm with Runge – Kutta and Newton – Raphson method to obtain velocity and temperature distributions. Besides, skin friction and Nusselt number are also computed for various physical parameters governing the problem under consideration. It is found that the inertial parameter has a significant influence in decreasing the flow field, whereas its influence is reversed on the rate of heat transfer for all values of permeability considered. The effect of Magnetic field is diminution with velocity of the fluid flow. Further, the obtained results under the two limiting conditions were found to be in good agreement with the existing results.

**Keywords:** Heat transfer, MHD, Newtonian fluid, porous medium, similarity solution.

[1] J H Merkin The effects of buoyancy forces on the boundary layer flow over a semi infinite vertical plate in a uniform free stream, J.

Fluid Mech., Vol.35, 1969, 439-450.

[2] P Ranganathann, R Viskanta, Mixed convection boundary layer flow along a vertical surface in a porous medium. Num Heat Transfr 7,
1984, 305-317.

[3] J.C. Hsieh, T.S. Chen, B.F.Armaly, Non-similarity solutions for mixed convection from vertical surfaces in porous medium with variable
surface temperature or heat flux, Int.J.Heat Mass Transfer, vol.38 ,No.4,1993,1485-1493.

[4] B. Saltzmann, The General Circulation as a problem in Thermal Convection: A Collection of Classical and Modern Theoretical Papers,
Dept. of Meteorology, Massachusetts Institute of Technology (1958)

[5] T.K.Aldoss, M.A. Al-Nimr, M.A.Jarrah, B.J.Al-shaer, Magnetohydrodynamic Mixed Convection from a vertical plate embedded in a
porous medium, Numerical Heat Transfer: An International Journal of Computation and Methodology. Vol.28, Issue 5. 1995,.635-645

[6] D.A.Nield and A. Bejan Convection in porous media.(Springer verlag 1999).

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**Abstract:**Oscillatory behaviors of second order forced functional differential equation is considered. The
oscillation of this equation is shown to be maintained under the effect of certain forcing terms, and the
oscillatory equation can serve as mathematical tool for simulation of processes and phenomina observed in
control theory.

**Key words:** Amplitude, functional differential equations, Damped oscillation, Oscillatory behaviors.

[1] Dass P. and Misra N. (1997) Necessary and sufficient conditions for the solution of a functional differential equations. To be

oscillatory. Journal of mathematical analysis and applications.

[2] Erbe L. H. and Qingkai K. H. (1995) oscillation theory for functional differential equations. Volume 190 of monographs and text

books in pure and applied mathematics. Marcel Decker Inc.New York.

[3] Hamadani G.G. (1995). Oscillatory behavior of nth order forced functional differential equations. Journal of mathematical analysis and
applications.

[4] Kreyszig E. (1999). Advanced engineering mathematics 8th edition.

[5] Lijun P. and Jinde C. (2010) Oscillation of even order linear implosive delay differential equations. Journal of differential equations

and applications. 2(2) 163-176.

[6] McGrew-Hill (1978). Dictionary of science and mathematics.

[7] Richard F. (2010) Simple harmonic oscillator equation. From Math World-a wolfram web resource. http/mathworld.wolfram.com.

applications.

[8] Tang X.H. (2002). Oscillation for first order supper linear delay differential equations. Journal, London mathematical society.

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**Abstract:** In a first order Markov process, if the state is known for any specific values of the time parameter,
that information is sufficient to predict the next behavior of the process beyond that point. This principle was
used to formulate a four-state model in continuous time to study the annual rainfall data with respect to the
annual rainfall distribution for crop production in minna. It was observed that if it is low rainfall in a given
year, it would take at most 25%,33% and 27% of the time to make a transition to moderate rainfall also well
spread, high rainfall, and moderate rainfall but not well spread respectively in the far future. Thus given the
rainfall in a year, it is possible to determine quantitatively the probability of finding rainfall in other states in
the following year and in the long run. This is an important information that could assist the farmers to plan
strategies for high crop production in the region.

**Keywords**: Markov Model, Transition probability, Annual rainfall, Crop production.

[1] Akintunde A.A, Asiribo O.E. Adebanji A.O. Adelakun A.A Agwuegbo S.O.N. (2008) Stochastic modelling of daily precipitation in Abeokuta, proceedings of the third Confersence on science and National Development pp108-118 www.unaab.edu.ng/journal/index.php/COLNAS/article/.../150/153. Accessed 15/10/2012.

[2] Tamil S. and Samuel S.(2011).Universal Journal of Environmental Research and Technology vol.1 Issue4 pp566-570.

[3] Howard R.A.(1960) Dynamic programming and Markov processes. The M.I.T.PRESS, Massachusetts.

[4] Korve K.N.(2000). A Three-State Continuous Time Markov Model for asthma process. ABACUS, The journal of Mathematical Association of Nigeria. Volume27, Number2, pp 33-46.

[5] Bhat,U.N. (1984) ''Element of Applied Stochastic processes'', John Wiley, New York.

[6] Okorie C. (2012). Semi-Markov Model in Discrete time and States and the Application to Desertification Study in Nigeria. Unpublished MSc. Thesis. Department of Mathematics and Statistics, Federal University of Technology, Minna, Nigeria.

[7] Korve K N(1994). Operations Research MAT537, Unpublished lecture notes. Department of Mathematics, University of Jos Nigeria

[8] Lawal Adamu(2012), The Use of Markov Model in Predicting the Rainfall Pattern forCropProduction. Unpublished MTech Thesis, Department of Mathematics and Statistics, Federal University of Technology,Minna. Nigeria

[9] Abubakar U.Y. and Lawal A. Muhammed A., (2013), Markov Chain Model and Its Application to annual Rainfall Distribution for Crop Production. Submitted.

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Paper Type |
: | Research Paper |

Title |
: | Compatible Mapping and Common Fixed Point Theorem |

Country |
: | India |

Authors |
: | Rajesh Shrivastava, Neha Jain, K. Qureshi |

: | 10.9790/5728-0714648 |

**Abstract:** In this paper we prove common fixed point theorem for compatible mapping.

**Key Words:** Common Fixed Point, Compatible Mapping ,Commuting mapping ,Metric Space

[1] Banach , S., Sur les operations dans les ensembles abstraits et leurs applications ,Fund. Math. 3:133-181(1922).

[2] Das , K.M. and Naik ,K.V., common Fixed point Theorems for commuting maps Mon a Metric Space, Proc.Amer. Math.Soc.77:369(1979).

[3] Fisher, B., common Fixed Point of commuting mappings, Bull. Inst. Math. Acad. Scinica , 9:399(1981).

[4] Fisher , B., common Fixed Point of four mappings, Bull. Inst.Math. Acad. Scinica ,11: 103(1983).

[5] Iseki , K. and Singh ,Bijendra ,On common Fixed point theorems of mappings Math. Sem. Notes,Kobe Univ., 2: 96(1974).

[6] Jungck, G., commuting maps and Fixed points Amer .Math. Monthly, 83: 261(1976). [7] Jungck, G., compatible mappings and common Fixed points ,Internat. J. Math. and Math .Sci., 9: 771(1986).

[8] ungck, G., compatible mappings and common Fixed points (2) , Internat. J. Math. and Math .Sci., 11: 285(1985).

[9] Kang , S.M. and Kim ,Y.P., Common Fixed point theorems, Math. Japonica , 37: 1037(1992).

[10] Khan ,M .S. and Imdad , M., Some common Fixed Point theorems, Glesnik Mat.18:(38), 321(1983).

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Paper Type |
: | Research Paper |

Title |
: | Perishable Inventory Model Having Weibull Lifetime and Time Dependent Demand |

Country |
: | India |

Authors |
: | Dr. R. John Mathew |

: | 10.9790/5728-0714954 |

**Abstract:** In this paper we develop and analyse an inventory model for deteriorating items with Weibull rate of decay and time dependent demand. Using the differential equations, the instantaneous state of inventory at time't', the amount of deterioration etc. are derived. With suitable cost considerations the total cost function and profit rate function are also obtained by maximizing the profit rate function, the optimal ordering and pricing policies of the model are derived. The sensitivity of the model with respect to the parameters is discussed through numerical illustration. It is observed that the deteriorating parameters have a tremendous influence on the optimal selling price and ordering quantity.

**Keywords: **Instantaneous rate of deterioration, Perishability, Profit rate function, the optimal ordering and pricing policies, Total cost function

[1] Goyal, S.K.,Giri, B.C (2001) Inview recent trends in modeling of deteriorating inventory, EJOR , Vol. 134, 1-16.

[2] Aggarwal, S.P., Goel, V.P (1984) Order Level inventory system with demand pattern for deteriorating items, Econ. Comp. Econ. Cybernet, Stud. Res.,Vol. 3,57- 69.

[3]. Giri, B.C., Goswami, A. and Chaudhuri, K. S. (1996). An EOQ model for deteriorating items with time varying demand and costs. Journal of the Operational Research Society, Vol.47, 1398-1405.

[4]. Mahata G. C. and Goswami A. (2009a) 'Fuzzy EOQ Models for Deteriorating Items with Stock Dependent Demand & Non-Linear Holding Costs', International Journal of Applied Mathematics and Computer Sciences 5;2, 94-98.

[5]. Manna, S.K., Chaudhuri, K.S. and Chiang, C. (2007) 'Replenishment policy for EOQ models with time-dependent quadratic demand and shortages', International journal of Operational Research, Vol. 2, No.3 pp. 321 – 337

[6] Mathew, R.J., Narayana, J.L. (2007) Perishable inventory model with finite rate of replenishment having weibull lifetimeand price dependent demand Assam Statistical review (2007), Vol. 21, 91-102.

[7] Mathew, R.J(2013) Perishable inventory model with finite rate of replenishment having weibull lifetime and time dependent demand .Accepted by International journal of mathematical archive,IJMR;4 – 228

[8]. Ritchie, E. (1984). The EOQ for linear increasing demand, A simple optimum solution. Journal of the Operational Research Society, Vol.35, 949-952.

[9]. Skouri, K., Konstantaras, I., Papachristos, S., Ganas, I., (2009) 'Inventory models with ramp type demand rate, partial backlogging and Weibull deterioration rate', European Journal of Operational Research, Vol. 192 (1), 79–92.

[10]. Dye C.Y., Hsieh T.P., and Ouyang L.Y. (2007) 'Determining optimal selling price and lot size with a varying rate of deterioration and exponential partial backlogging', European Journal of Operational Research, Vol. 181(2), 668-678

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**Abstract:**This paper conducts an application of the DEA Methodology in the assessment of the performance of JNTUH Colleges the indicators included the Faculty, Students, Infrastructure and Placements of the technical Institutions. The results reveal those institutions that more efficiently carry out these activities. The proposed method has been used for selection of quality attributes in technical education setting the performance of an institute is likely to be influenced by quality of teacher, quality of students, infrastructure administration, extent of training and placement and many others. It is felt that quality and performance evaluation is necessary not only for appraisal but it is also required to improve overall service quality. Finally we discuss about the existence of differences in the strengths and weaknesses between the technical institutions.

**Key Words: **BCC Model, Data Envelopment Analysis, Efficiency Measurement Returns to Scale

[1] I-Huei Ho, Tzai-Fu Cheng and Chieh-Yu Lin, (2001), "The Construction of the Performance Evaluation Model for Engineering Education Systems", International Conference on Engineering Education, August 6-10, Oslo, Norway.

[2] Ana Lúcia Miranda Lopes and Edgar Augusto Lanzer, (2002), "Data Envelopment Analysis-DEA and Fuzzy Sets to Assess the Performance of an Academic Departments: A Case Study at a Federal University of Santa Catarina-UFSC", Pesquisa Operacional,Volume 22, Number 2, pp. 217-230.13

[3] Emilio Martin, (2003), "An Application of the Data Envelopment Analysis Methodology in the Performance Assessment of the Zaragoza University Departments", Documento deTrabajo 2003-06.

[4] John Ruggiero, (2004), "Performance Evaluation When Non-Discretionary Factors Correlate with Technical Efficiency", European Journal of Operations Research, Volume159, pp. 250-257.

[5] Hahn-Ming Lee, Chi-Chun Huang and Tzu-Ting Kao, (2005), "Personalized Course Navigation Based on Grey Relational Analysis", Applied Intelligence, Volume 22, pp.83-92.

[6] Kosmas Kotivas, Maria L. Tsipa and Dimitrios N. Tsipas, (2005), "The Development and Implementation of a Pilot Internal Evaluation Procedure for Postgraduate Engineering Courses", World Transactions on Engineering & Technical Education, Volume 4, Number 1, pp. 83-86.

[7] P. Kousalya, V. Ravindranath and K. Vizayakumar, (2006), "Student Absenteeism in Engineering Colleges: Evaluation of Alternatives Using AHP", Journal of Applied Mathematics and Decision Sciences, Volume 2006, Article ID 58232, pp. 1-26.

[8] Salah-Ud-Din Khan, Mearaj-Ud-Din and Muhammad Shah, (2006), "Development of aReliable and Valid Instrument for the Evaluation of Performance of Directors of Physical Education", Gomal University Journal of Research, Volume 22, pp. 29-26.

[9] Cai Yonghong and Lin Chongde, (2006), "Theory and Practice on Teacher Performance Evaluation", Front. Education China, Volume 1, pp. 29-39.

[10] Mary Caroline N. Castano and Emilyn Cabanda, (2007), "Sources of Efficiency and Productivity Growth in the Philippine State Universities and Colleges: A Non-Parametric Approach", International Business and Economics Research Journal, Volume 6, Number 6, pp. 79-90.

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Paper Type |
: | Research Paper |

Title |
: | ON M(M,m)/M/C/N: Interdependent Queueing Model |

Country |
: | India |

Authors |
: | R. John Mathew, Varaprasad B. Sabbithi, J. Lakshinarayana |

: | 10.9790/5728-0715962 |

**Abstract:** This paper deals with a multiple server Queueing system in which arrivals and services are
intredependent and follow a bivariate Poisson process and having startup delay. Using the Supplementary
Varible Techniue these models are analyzed. The expected length of the Dorment period, the busy period,
expected number of units in the queue are derived and analyzed in the light of the dependence paramater.

**Keywords: **interdependent queue, bivariate Poisson process, the length of the Dorment period, the busy period,
expected number of units in the queue

[1] Aftab Begum, et.al, "The M/M/C Interdependent Queueing model with controllable arrival rates". Opsearch,39(2),pp89 -110(2002).

[2] Bhat, U.N, "Queueing Systems with First Order Dependence", OPSEARCH..(1969)

[3] Conolly, W.B.and Hadidi, N. A, "Correlated Queue", J.Appl. Prob., Vol.5. (1969)

[4] Mathew.R.J et.al, "On Mn/M/C/N: Interdependent Queueing Model". International Journal of Management And Systems, Vol.14,

No.2, pp 167-176, (1998).

[5] Maurya V.N,"On the expected busy period of an interdependent M/M/1:(,Gd) Queueing model using Bi-variate Poisson process

and controllable arrival rates", IEEE Trans. pp.243-246(2010)

[6] Mishra S.S. ,"Optimam performance measues of interdependent queueing system with controllable arrival rates", Int, Journ.

Mathematical, Physical and engineering sciences pp.72-75(2009).

[7] Prasad Reddy and K Srinivasa Rao, et.al, "Interdependent Queueing model with jockeying", Ultra science, Vol.18 (1) M, pp.87 -

98(2006).

[8] Rao, K.S. "ON an interdependent communication networks"Opsearch,37(2),pp,134-143(2000).

[9] Simha, P.S. "Optimal Operating Policies for the finite Source Queueing Models", Ph.D. Thesis, Delhi University, India. (1971)

[10] Soble. M.J. "optimal average cost policy for a queue with startup and shutdown costs", Operations Research, Vol.17. (1969)

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**Abstract:** The aim of this paper is to introduce the concept of fuzzy - semi open and fuzzy - semi closed sets of
a fuzzy topological space. Some characterizations are discussed, examples are given and properties are
established. Also, we define fuzzy - semi interior and fuzzy - semi closure operators. And we introduce fuzzy
- t-set, -SO extremely disconnected space analyse the relations between them.
MSC 2010: 54A40, 03E72.

**Key words:** Fuzzy - open, fuzzy - closed, fuzzy - semi open, fuzzy - semi closed, fuzzy - semi interior and
fuzzy - semi closure, - t-set and fuzzy topology.

[1] M.Athar and B.Ahmad, Fuzzy Boundary and Fuzzy Semiboundary, Advances in Fuzzy systems, vol. 2008, 586893, 9 pages.

[2] K.K.Azad, On fuzzy semi-continuity, fuzzy almost continuity and fuzzy weakly continuity, J.Math.Anal.Appl. 82(1)(1981), 14-32. [3] Biljana Krsteska , Fuzzy Strongly preopen sets and fuzzy strongly precontinuity, Mat. Vesnik 50(1998),111-123.

[4] C.L.Chang, Fuzzy Topological Spaces, J.Math. Anal, Appl.24(1968), 182-190.

[5] Luay A.Al.Swidi, Amed S.A.Oon, Fuzzy -open sets and fuzzy -closed sets, Americal Journal of scientific research, 27(2011), 62-67.

[6] T.Noiri and O.R.Sayed, Fuzzy -open sets and fuzzy -continuity in fuzzitying topology, Sci. Math. Jpn. 55(2002).

[7] A.K. Katsaras and D.B. Liu, Fuzzy vector spaces and fuzzy topological vector spaces , .Math.Anal.Appl., vol. 58,no 1, pp. 135-146, 1977. [8] L.A.Zadch, Fuzzy Sets, Information and control, vol.8, Pp 338-353, 1965.