#### Volume-3 ~ Issue-3

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**Abstract :**The present paper considers a more practical problem of scheduling n jobs in a two machine specially structured open shop to minimize the rental cost. Further the processing time of jobs is associated with their respective probabilities including transportation time. In most of literature the processing times are always considered to be random, but there are significant situations in which processing times are not merely random but bear a well defined structural relationship to one another. The objective of this paper is to minimize the rental cost of machines under a specified rental policy. The algorithm is demonstrated through the numerical illustration.

**Keywords -**Open Shop Scheduling, Rental Policy, Processing Time, Utilization Time, Make span, Idle Time. Mathematical Subject Classification: 90B30, 90B35

[1] Anup (2002), "On two machine flow shop problem in which processing time assumes probabilities and there exists equivalent for

an ordered job block", JISSOR, Vol. XXIII No. 1-4, pp. 41-44.

[2] Bagga P C (1969), "Sequencing in a rental situation", Journal of Canadian Operation Research Society, Vol.7, pp.152-153.

[3] Baker, K. R. (1974), "Introduction of sequencing and scheduling," John Wiley and Sons, New York.

[4] Bellman, R. (1956), "Mathematical aspects of scheduling theory", J. Soc. Indust. Appl. Math. 4(3),168-205.

[5] Belwal & Mittal (2008), "n jobs machine flow shop scheduling problem with break down of machines,transportation time and

equivalent job block", Bulletin of Pure & Applied Sciences-Mathematics, Jan – June,2008, source Vol. 27, Source Issue 1.

[6] Chander S, K Rajendra & Deepak C (1992), "An Efficient Heuristic Approach to the scheduling of jobs in a flow shop",

European Journal of Operation Research, Vol. 61, pp.318-325.

[7] Chandramouli, A. B. (2005), "Heuristic approach for n-jobs, 3-machines flow-shop scheduling problem involving transportation

time, breakdown time and weights of jobs", Mathematical and ComputationalApplications 10(2), pp 301-305.

[8] Chandrasekharan R (1992), "Two-Stage Flowshop Scheduling Problem with Bicriteria " O.R. Soc. ,Vol. 43, No. 9, pp.871-84.

[9] D. Rebaine, V.A. Strusevich(1998), Two-machine open shop scheduling with special transportation times, CASSM R&D Paper 15,

University of Greenwich, London, UK..

[10] Gupta Deepak (2005), "Minimizing rental cost in two stage flow shop , the processing time associated with probabilities including

job block", Reflections de ERA, Vol 1, No.2, pp.107-120.

an ordered job block", JISSOR, Vol. XXIII No. 1-4, pp. 41-44.

[2] Bagga P C (1969), "Sequencing in a rental situation", Journal of Canadian Operation Research Society, Vol.7, pp.152-153.

[3] Baker, K. R. (1974), "Introduction of sequencing and scheduling," John Wiley and Sons, New York.

[4] Bellman, R. (1956), "Mathematical aspects of scheduling theory", J. Soc. Indust. Appl. Math. 4(3),168-205.

[5] Belwal & Mittal (2008), "n jobs machine flow shop scheduling problem with break down of machines,transportation time and

equivalent job block", Bulletin of Pure & Applied Sciences-Mathematics, Jan – June,2008, source Vol. 27, Source Issue 1.

[6] Chander S, K Rajendra & Deepak C (1992), "An Efficient Heuristic Approach to the scheduling of jobs in a flow shop",

European Journal of Operation Research, Vol. 61, pp.318-325.

[7] Chandramouli, A. B. (2005), "Heuristic approach for n-jobs, 3-machines flow-shop scheduling problem involving transportation

time, breakdown time and weights of jobs", Mathematical and ComputationalApplications 10(2), pp 301-305.

[8] Chandrasekharan R (1992), "Two-Stage Flowshop Scheduling Problem with Bicriteria " O.R. Soc. ,Vol. 43, No. 9, pp.871-84.

[9] D. Rebaine, V.A. Strusevich(1998), Two-machine open shop scheduling with special transportation times, CASSM R&D Paper 15,

University of Greenwich, London, UK..

[10] Gupta Deepak (2005), "Minimizing rental cost in two stage flow shop , the processing time associated with probabilities including

job block", Reflections de ERA, Vol 1, No.2, pp.107-120.

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**Abstract :**This research deals with mathematical modelling of malaria transmission in North Senatorial Zone of Taraba State, Nigeria. The SIR proposed by Kermack and McKendrick and data obtained from Essential Programme on Immunisation (EPI) unit, F.M.C., Jalingo, Taraba state were used to analyse the rate of infection of malaria in the zone. From our analysis, we found out that the reproduction ratio (R0 ) 0 . Based on the reproduction ratio 0 R , which is greater than 0, implies that the force of malaria infection in Taraba North Senatorial Zone is high. The researchers also make recommendations for the reduction of malaria in the zone. ,br>

**Keywords:**Malaria, stability, equilibrium states, epidemics.

[1] Anderson, R. M., May, R. M. (1991). Infectious Diseases of Humans: Dynamics and Control.Oxford University Press, Oxford.

[2] Hyun, M. Y. (2000). Malaria transmission model for different levels of acquired immunity and temperature dependent parameters

vector. Rev. Saude Publica., 34(3): 223-231.

[3] Isao, K., Akira, S., Motoyoshi, M. (2004). Combining Zooprophylaxis and insecticide spraying A malaria-control strategy limiting

the development of insecticides resistance in vector mosquitoes. Proc. R. Soc. Lond., 271: 301-309.

[4] Jia, L. (2008). A malaria model with partial immunity in humans. Math. Bios. Eng., 5(4): 789- 801.

[5] Kermack, W. O. and Mckendric, A.G. ( 1927). A contribution to the mathematical theory of epidemics: preceedings of the Royal

society of London. Series A, Containing papers of a mathematical and physical character, 115:700-721.

[6] Makinde, O. D., Okosun, K. O. (2011). Impact of chemo-theraphy on optimal control of malaria disease with infected immigrants.

BioSystems. 104:32-41

[7] McDonald, G. (1957) The epidemiology control of malaria, Oxford university press, London.

[8] Okosun, K. O. (2010). Mathematical epidemiology of Malaria Disease Transmission and its Optimal Control Analyses, Ph.D.

thesis, University of the Western Cape, South Africa Puntani, P. I-ming, T. (2010). Impact of cross-border migration on disease

epidemics: case of the P. falciparum and P. vivax malaria epidemic along the Thai-Myanmar border. J. Bio. Sys., 18(1): 55-73

[9] Rafikov, M., Bevilacqua, L., Wyse, A. A. P. (2009). Optimal control strategy of malaria vector using genetically modified

mosquitoes. J. Theore. Bio., 258: 418-425.

[10] RollbackMalaria.What is Malaria?http://woo.rollback-malaria.org/cmc- upload/0/000/015/372/RBMinfosheet-1.pdf.(2010-05-

10).

[2] Hyun, M. Y. (2000). Malaria transmission model for different levels of acquired immunity and temperature dependent parameters

vector. Rev. Saude Publica., 34(3): 223-231.

[3] Isao, K., Akira, S., Motoyoshi, M. (2004). Combining Zooprophylaxis and insecticide spraying A malaria-control strategy limiting

the development of insecticides resistance in vector mosquitoes. Proc. R. Soc. Lond., 271: 301-309.

[4] Jia, L. (2008). A malaria model with partial immunity in humans. Math. Bios. Eng., 5(4): 789- 801.

[5] Kermack, W. O. and Mckendric, A.G. ( 1927). A contribution to the mathematical theory of epidemics: preceedings of the Royal

society of London. Series A, Containing papers of a mathematical and physical character, 115:700-721.

[6] Makinde, O. D., Okosun, K. O. (2011). Impact of chemo-theraphy on optimal control of malaria disease with infected immigrants.

BioSystems. 104:32-41

[7] McDonald, G. (1957) The epidemiology control of malaria, Oxford university press, London.

[8] Okosun, K. O. (2010). Mathematical epidemiology of Malaria Disease Transmission and its Optimal Control Analyses, Ph.D.

thesis, University of the Western Cape, South Africa Puntani, P. I-ming, T. (2010). Impact of cross-border migration on disease

epidemics: case of the P. falciparum and P. vivax malaria epidemic along the Thai-Myanmar border. J. Bio. Sys., 18(1): 55-73

[9] Rafikov, M., Bevilacqua, L., Wyse, A. A. P. (2009). Optimal control strategy of malaria vector using genetically modified

mosquitoes. J. Theore. Bio., 258: 418-425.

[10] RollbackMalaria.What is Malaria?http://woo.rollback-malaria.org/cmc- upload/0/000/015/372/RBMinfosheet-1.pdf.(2010-05-

10).

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**Abstract :**In this work, The researchers present Passively Immune Infant( ) m V -Susceptible class( ) m S - Infection class( ) m I -Recovery class( ) m R model to study the dynamic of tuberculosis transmission and vaccination impact in North Senatorial Zone, Taraba State, Nigeria. The compartment of the model is presented in a system of ordinary differential equations. Quantitative analysis of the model was done to investigate the equilibrium and stability of the model. An analytical approach was used to determine their Disease Free equilibrium and the Epidemic equilibrium state. The stability of the epidemic equilibrium is tested using Bellman and Cooke's theorem. The model had two equilibrium position: The disease free equilibrium which was asymptotically stable for ( ) 0 e R and the endemic equilibrium which was locally asymptotically stable for as it satisfies the Bellman and Cooke's condition for stability i.e. J 0 .

**Keywords -**Tuberculosis (TB), Vaccination, Infection, Equilibrium analysis, Stability analysis.

[1] Aparicio, J.P., Capurri, A.F., Castillo-Chavez, C. (2000), "Transmission and dynamics of Tuberculosis on generalized household',

J. Theoretical Biology.

[2] Bellman R. and Cooke K.C. (1963), "Differential difference equation". London. Academic press.

[3] Castillo-Chavez, C. and Feng, Z., (1998), Global stability of an age-structure model for TB and its applications to optimal

vaccination strategies. Math Biosci,151(2):135–154.

[4] Dye, C., Garnett, G. P., Sleeman, K. and Williams, B. G.(1998), Prospects for Worldwide tuberculosis control under the who dots

strategy. Directly observed short-course therapy.

[5] Fine PEM (1988). BCG vaccination against tuberculosis and leprosy.Br Med Bull; 44:691-703.

[6] Fine PEM, Ponnighaus J.M., Maine N.P. (1986), The relationship between delayed type hypersensitivity and protective immunity

induced by mycobacterial vaccines in man.Symposium on the Immunology of Leprosy, Oslo, Norway.

[7] Heimbeck J. Sur la vaccination préventive de la tuberculose par injection sous-cutanée de BCG chez les élèves infirmières de

l'hôpital Ulleval à Oslo (Norvège).

[8] Milstien JB, Gibson JJ. (1989)., Quality control of BCG vaccines by the World Health Organization: a review of factors that may

influence vaccine effectiveness and safety. Bull WHO; 68:93-108.

[9] Myint TT, Win H, Aye HI-I, (1987), Case-control study on evaluation of BCG vaccination of newborn in Rangoon, Burma. Ann

Trop Paediatr 1987;7:159-166.

[10] Ndaman I.(2010) A deterministic mathematical model of Tuberculosis disease dynamics, M. TECH thesis, F.U.T Minna.

J. Theoretical Biology.

[2] Bellman R. and Cooke K.C. (1963), "Differential difference equation". London. Academic press.

[3] Castillo-Chavez, C. and Feng, Z., (1998), Global stability of an age-structure model for TB and its applications to optimal

vaccination strategies. Math Biosci,151(2):135–154.

[4] Dye, C., Garnett, G. P., Sleeman, K. and Williams, B. G.(1998), Prospects for Worldwide tuberculosis control under the who dots

strategy. Directly observed short-course therapy.

[5] Fine PEM (1988). BCG vaccination against tuberculosis and leprosy.Br Med Bull; 44:691-703.

[6] Fine PEM, Ponnighaus J.M., Maine N.P. (1986), The relationship between delayed type hypersensitivity and protective immunity

induced by mycobacterial vaccines in man.Symposium on the Immunology of Leprosy, Oslo, Norway.

[7] Heimbeck J. Sur la vaccination préventive de la tuberculose par injection sous-cutanée de BCG chez les élèves infirmières de

l'hôpital Ulleval à Oslo (Norvège).

[8] Milstien JB, Gibson JJ. (1989)., Quality control of BCG vaccines by the World Health Organization: a review of factors that may

influence vaccine effectiveness and safety. Bull WHO; 68:93-108.

[9] Myint TT, Win H, Aye HI-I, (1987), Case-control study on evaluation of BCG vaccination of newborn in Rangoon, Burma. Ann

Trop Paediatr 1987;7:159-166.

[10] Ndaman I.(2010) A deterministic mathematical model of Tuberculosis disease dynamics, M. TECH thesis, F.U.T Minna.

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Paper Type |
: | Research Paper |

Title |
: | Differential Equations in Stability Analysis of Ferrofluids |

Country |
: | India |

Authors |
: | Dr.R.Vasanthkumari , A.Selvaraj |

: | 10.9790/5728-0332427 |

**Abstract :**The application of differential equations towards stability analysis of ferrofluids is analysis both in porous medium and nonporous medium and a comparative analysis is made. Weakly non- linear analysis is carried out. A mathematics model of the differential equations employed is presented. The non-dimensional thermal Rayleigh number Ra and magnetic Rayleigh number Rm are analysed with allowable range of parameter.

**Keywords -**Ferrofluids, Mathematical Model, Non Porous Medium, Porous Medium,Weakly Non Linear Equations.

[1]. Baily R.L (1983): Lessor known applications of ferrofluids:- J.M.M.M.,vol.39,pp.178-182.

[2]. Berkovskii B.M., Medvedev V.F. and Krakov M.S(1993): Magnetic fluids-engineering applications-oxford: oxford science

publications.

[3]. Chandrasekher S.(1961): Hydrodynamic and stability-oxford:Clarendon.

[4]. Finlayson B.A. (1970): Convective instability of ferromagnetic fluids –Journal of Fluid mech., Vol.40,pp.753-767.

[5]. E.R.Benton(1966). On the flow due to a rotating disk- Journal of Fluid mech. 24(4),pp.781-800.

[6]. H.Schlichting (1960), Boundary Layer Theory, McGraw-Hill Book company, New York..

[7]. H.A.Attia (2009),Steady flow over a rotating disk in porous medium with heat transfer. Non-Linear analysis modelling and

control.14(1)pp.21-26.

[8]. J.L.Newringer, R.E.Rosensweig(1964),Magnetic fluids, Physics of fluids ,1927.

[9]. M.I.Shliomis(2004), Ferrofluids as thermal ratchets. Physical Review Letters,92(18),188901.

[10]. P.D.S. Verma, M.Singh (1981),Magnetic fluid flow through porous annulus. Int.J.Non-linear Mechanics,16(3/4),pp.371-378.

[2]. Berkovskii B.M., Medvedev V.F. and Krakov M.S(1993): Magnetic fluids-engineering applications-oxford: oxford science

publications.

[3]. Chandrasekher S.(1961): Hydrodynamic and stability-oxford:Clarendon.

[4]. Finlayson B.A. (1970): Convective instability of ferromagnetic fluids –Journal of Fluid mech., Vol.40,pp.753-767.

[5]. E.R.Benton(1966). On the flow due to a rotating disk- Journal of Fluid mech. 24(4),pp.781-800.

[6]. H.Schlichting (1960), Boundary Layer Theory, McGraw-Hill Book company, New York..

[7]. H.A.Attia (2009),Steady flow over a rotating disk in porous medium with heat transfer. Non-Linear analysis modelling and

control.14(1)pp.21-26.

[8]. J.L.Newringer, R.E.Rosensweig(1964),Magnetic fluids, Physics of fluids ,1927.

[9]. M.I.Shliomis(2004), Ferrofluids as thermal ratchets. Physical Review Letters,92(18),188901.

[10]. P.D.S. Verma, M.Singh (1981),Magnetic fluid flow through porous annulus. Int.J.Non-linear Mechanics,16(3/4),pp.371-378.

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Paper Type |
: | Research Paper |

Title |
: | Some properties of Fuzzy Derivative (I) |

Country |
: | Iraq |

Authors |
: | Salah Mahdi Ali |

: | 10.9790/5728-0332829 |

**Abstract :**In [3], the fuzzy derivative was defined by using Caratheodory's derivative notion and a few basic properties of fuzzy derivative was proved. In this paper, we will a completion to prove for some properties of the subject and discussion Rolle's theorem and Generalized Mean -Value Theorem in fuzzy derivative and we given some applications of the Mean Value Theorem

**Keywords -**Chain Rule, Critical point theorem, Rolle's Theorem, Generalized Mean – Value Theorem. MSC (2010): 54A40,46S40

[1] T. M. Apostol , Mathematical Analysis, Addison-Wesley Pub. Com., Inc., (1964) .

[2] R. M. Dudley, Real Analysis and Probability, Cambridge Univ. press,(2004).

[3] S. Fuhua, The Basic Properties of Fuzzy Derivative, BUSEFAL, 76(1998),120-123.

[4] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, Inc., (1976).

[5] A. K. Katsaras, Fuzzy Topological Vector Spaces I, Fuzzy Sets and Systems, 6 (1981) 85-95 .

[2] R. M. Dudley, Real Analysis and Probability, Cambridge Univ. press,(2004).

[3] S. Fuhua, The Basic Properties of Fuzzy Derivative, BUSEFAL, 76(1998),120-123.

[4] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, Inc., (1976).

[5] A. K. Katsaras, Fuzzy Topological Vector Spaces I, Fuzzy Sets and Systems, 6 (1981) 85-95 .

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Paper Type |
: | Research Paper |

Title |
: | Single Variable Unconstrained Optimization Techniques Using Interval Analysis |

Country |
: | India |

Authors |
: | G. Veeramalai, R.J.Sundararaj |

: | 10.9790/5728-0333034 |

**Abstract :**In this paper, we discussed single variable unconstrained optimization techniques using Interval Analysis. The most of the unconstrained linear problems have been dealt with differential calculus methods. But, here non-linear unconstrained problems are solved using Newton's method by establishing Interval Analysis method. Establishing Interval Analysis method gives more accurate root even for higher order derivatives.

**Keywords -**Interval analysis, Newton's method, Single variable, unconstrained optimization techniques

[1] E. Hansan and G. W. Walster, "Global optimization using Interval Analysis", Marcel Dekker, New York, 2003.

[2] Karl Nickel, On the Newton method in Interval Analysis. Technical report 1136, Mathematical Research Center, University of

Wisconsion, Dec1971

[3] Helmut Ratschek and Jon G. Rlkne, New Computer Methods for Global Optimization, Wiley, New York, 1988.

[4] Louis B. Rall, A Theory of interval iteration, proceeding of the American Mathematics Society, 86z:625-631, 1982.

[5] Louis B. Rall, Application of interval integration to the solution of integral equations. Journal of Integral equations 6: 127-

141,1984.

[6] Ramon E. Moore, R. Baker Kearfoth, Michael J. Cloud, Introduction to interval analysis, SIAM, 105-127, Philadelphia, 2009.

[7] Eldon Hansen, Global optimization using interval analysis- Marcel Dekker, 1992

[8] Hansen E.R (1978a), " Interval forms of Newton‟s method, Computing 20, 153-163.

[9] Hansen E. R (1979), "Global optimization using interval analysis-the one dimensional case, J.Optim, Theory Application, 29, 314-

331.

[10] Hansen E. R (1988), An overview of Global Optimization using interval analysis in Moore(1988) pp 289-307.

[2] Karl Nickel, On the Newton method in Interval Analysis. Technical report 1136, Mathematical Research Center, University of

Wisconsion, Dec1971

[3] Helmut Ratschek and Jon G. Rlkne, New Computer Methods for Global Optimization, Wiley, New York, 1988.

[4] Louis B. Rall, A Theory of interval iteration, proceeding of the American Mathematics Society, 86z:625-631, 1982.

[5] Louis B. Rall, Application of interval integration to the solution of integral equations. Journal of Integral equations 6: 127-

141,1984.

[6] Ramon E. Moore, R. Baker Kearfoth, Michael J. Cloud, Introduction to interval analysis, SIAM, 105-127, Philadelphia, 2009.

[7] Eldon Hansen, Global optimization using interval analysis- Marcel Dekker, 1992

[8] Hansen E.R (1978a), " Interval forms of Newton‟s method, Computing 20, 153-163.

[9] Hansen E. R (1979), "Global optimization using interval analysis-the one dimensional case, J.Optim, Theory Application, 29, 314-

331.

[10] Hansen E. R (1988), An overview of Global Optimization using interval analysis in Moore(1988) pp 289-307.

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Paper Type |
: | Research Paper |

Title |
: | On Semiopen Sets and Semicontinuous Functions in Intuitionistic Fuzzy Topological Spaces |

Country |
: | India |

Authors |
: | Shyamal Debnath |

: | 10.9790/5728-0333538 |

**Abstract :**The purpose of this paper is to introduce "semiopen sets" in intuitionistic fuzzy topological spaces. After giving the fundamental definitions and necessary examples we introduce the definitions of intuitionistic fuzzy semicontinuity, intuitionistic fuzzy semicompactness, intuitionistic fuzzy semiconnectedness and studied several preservations properties and some characterizations theorems. We see that every intuitionistic fuzzy open set is intuitionistic fuzzy semiopen and every intuitionistic fuzzy continuous function is intuitionistic fuzzy semicontinuous.

**Keywords -**Intuitionistic fuzzy topology, intuitionistic fuzzy semiopen sets, intuitionistic fuzzy semicontinuous functions, intuitionistic fuzzy semi 𝐶5 −connectedness, intuitionistic fuzzy semicompactness.

[1] K. Atanassov: Intuitionistic fuzzy sets, VII ITKR's Session, Sofia,1983.(In Bulgarian)

[2] K. Atanassov: Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1986), 87-96.

[3] K.K.Azad: On fuzzy semicontinuity, fuzzy almost continuity and fuzzy weakly continuity, J.Math.Anal.Appl.82(1981),14-32.

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

[5] A.K.Chaudhuri and P.Das, Fuzzy connected sets in fuzzy topological spaces, Fuzzy Sets and Systems 49(1992), 223-229.

[6] D.Coker and A.H.Es: On fuzzy compactness in intuitionistic fuzzy topological spaces, J. Fuzzy Math. 3(1995), 899-909.

[7] D.Coker: An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems 88(1997), 81-89.

[8] H.Gurcay, D.Coker and A.H.Es: On fuzzy continuity in intuitionistic fuzzy topological spaces, J.Fuzzy Math. 5(1997),365-378.

[9] I.M.Hanafy: Completely continuous functions in intuitionistic fuzzy topological spaces, Czechoslovak Mathematical Journal,

53(128) (2003), 793-803.

[10] L.A.Zadeh, Fuzzy sets, Inform. And Control 8(1965)338-353.

[2] K. Atanassov: Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1986), 87-96.

[3] K.K.Azad: On fuzzy semicontinuity, fuzzy almost continuity and fuzzy weakly continuity, J.Math.Anal.Appl.82(1981),14-32.

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

[5] A.K.Chaudhuri and P.Das, Fuzzy connected sets in fuzzy topological spaces, Fuzzy Sets and Systems 49(1992), 223-229.

[6] D.Coker and A.H.Es: On fuzzy compactness in intuitionistic fuzzy topological spaces, J. Fuzzy Math. 3(1995), 899-909.

[7] D.Coker: An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems 88(1997), 81-89.

[8] H.Gurcay, D.Coker and A.H.Es: On fuzzy continuity in intuitionistic fuzzy topological spaces, J.Fuzzy Math. 5(1997),365-378.

[9] I.M.Hanafy: Completely continuous functions in intuitionistic fuzzy topological spaces, Czechoslovak Mathematical Journal,

53(128) (2003), 793-803.

[10] L.A.Zadeh, Fuzzy sets, Inform. And Control 8(1965)338-353.

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Paper Type |
: | Research Paper |

Title |
: | Common Random Fixed Point theorem for compatible random multivalued operators |

Country |
: | India |

Authors |
: | Dr. Neetu Vishwakarma |

: | 10.9790/5728-0333943 |

**Abstract :**The aim of this paper is to prove some common random fixed point theorem for two pairs of compatible random multivalued operators satisfying rational inequality

**Keywords -**Random fixed point, Compatible maps, Polish space. AMS Mathematics Subject Classification (2000): 47H10, 54H25.

[1]. Badshah, V.H. and Sayyed, Farkhunda, Common random fixed point of random multivalued operator on Polish spaces, Indian J.

Pure App. Math. 33(4), Apr. 2002, 573-582.

[2]. Beg, I. Random fixed point of random operators satisfying semi-contractivity conditions, Mathematica Japonica 46(1997), no. 1,

151-155.

[3]. Beg, I., Approximation of random fixed point in normed spaces, Nonlinear Analysis, 51 (2002), No. 8, 1363-1372.

[4]. Beg, I. and Abbas, Mujahid, Common random fixed point of compatible random operator, Int. J. Math. and Math. Sci. Vol. 2006

Article I.D. 23486, 1-15.

[5]. Beg, I, and Shahzad, N., Random fixed points of random multivalued operators on Polish spaces, Non-linear Analysis Theory

Methods and Applications 20 (1993), 835-847.

[6]. Bharucha Reid, A.T. a. Random integral equations, Academic Press, New York, 1972.

[7]. Hans, O., Reduzierende zufallige transformation. Czechoslovak Mathematics Journal 7 (1957), 154-158.

[8]. Hans, O., Random operator equations, Proceeding of the 4 th Berkeley Symposium in Mathematical Statistics and Probability, Vol.

II, Part I, University of California Press, California (1961), 185-202.

[9]. Itoh, S., A random fixed point theorem for a multivalued contraction mapping, Pacific Journal Mathematics 68 (1977), 85-90.

[10]. Mukherjee, A., Random transformations of Banach spaces. Ph.D. Dissertation, Wayne State University Detroit, Michegall, USA

(1968.).

Pure App. Math. 33(4), Apr. 2002, 573-582.

[2]. Beg, I. Random fixed point of random operators satisfying semi-contractivity conditions, Mathematica Japonica 46(1997), no. 1,

151-155.

[3]. Beg, I., Approximation of random fixed point in normed spaces, Nonlinear Analysis, 51 (2002), No. 8, 1363-1372.

[4]. Beg, I. and Abbas, Mujahid, Common random fixed point of compatible random operator, Int. J. Math. and Math. Sci. Vol. 2006

Article I.D. 23486, 1-15.

[5]. Beg, I, and Shahzad, N., Random fixed points of random multivalued operators on Polish spaces, Non-linear Analysis Theory

Methods and Applications 20 (1993), 835-847.

[6]. Bharucha Reid, A.T. a. Random integral equations, Academic Press, New York, 1972.

[7]. Hans, O., Reduzierende zufallige transformation. Czechoslovak Mathematics Journal 7 (1957), 154-158.

[8]. Hans, O., Random operator equations, Proceeding of the 4 th Berkeley Symposium in Mathematical Statistics and Probability, Vol.

II, Part I, University of California Press, California (1961), 185-202.

[9]. Itoh, S., A random fixed point theorem for a multivalued contraction mapping, Pacific Journal Mathematics 68 (1977), 85-90.

[10]. Mukherjee, A., Random transformations of Banach spaces. Ph.D. Dissertation, Wayne State University Detroit, Michegall, USA

(1968.).

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**Abstract :**We discuss the T.O.M (Term Omission Method) to estimate the distribution parameters that have not single exponential family of exponential family with one single distribution parameter, and compare it with different methods using Mean Square Method (MSE).

**Keywords -**Exponential families, Exponential form,

[1] Andreas V. (2010), "Notes on exponential family distributions and generalized linear models".

[2] Watkins, Joseph C. (2009). "Exponential Families of Random Variables", University of Arizona, p: 1.

[3] Jurgen, S. (1999). "Mathematical Statistics I", Utah State University, p: 120.

[4] Robert V. Hogg and Allen T. Craig (1978). "Introduction to Mathematical Statistics", 4th Edition, p: 357.

[5] Anirban DasGupta A. (2011). "Probability for Statistics and Machine Learning", Springer, pp: 583-596.

[6] Lawrence D. Brown (1986). "Fundamentals of statistical exponential families with applications in

statistical decision theory", Vol. 9, IMS Series.

[7] Labban, J.A. (2012). "Estimation of Single Distributions Parameter by T.O.M with Exponential

Families", American Journals Science, Issue 55, pp. 70-75.

[8] Nicu S. and Michael S. Lew (2003). "Robust Computer Vision - Theory and Applications", Kluwer

Academic Publishers, Volume 26, p. 54.

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