#### Volume-9 ~ Issue-4

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**Abstract:** This paper is devoted to introduce the notion of fuzzy supra semi ̃ =0, 1, 2 space, fuzzy supra semi Di =0, 1, 2 space, and use the notion of fuzzy quasi coincident in their definitions, study some properties and theorems related to these subjects.

**Key words:** Fuzzy supra semi open set, fuzzy supra semi D set, fuzzy supra semi ̃ = 0, 1, 2 space, fuzzy supra semi Di =0, 1, 2 space.

[1] Abd EL-Monsef M.E. and Ramadan A. E. "on fuzzy supra topological spaces" J. Pure app. Math., 18(4):322-329(1987).

[2] Chakrabarty, M.K. and Ahsanullal,T.M.G." Fuzzy topology on fuzzy sets And Tolerance Topology" Fuzzy sets and systems 45,103-108(1992).

[3] Ghanim M.H, Tantawy O.A. and SelimFawziaM."Gradation of supra-openness", Fuzzy sets and Systems, 109, 245-250, (2000).

[4] HoqueMd F., and Ali D.M "Supra fuzzy Topological space" Lap Lambert Academic Publishing GmbH and Co. KG, (2012).

[5] Kandil , A. and El-Shafee, M. E., "Regularity Axioms in Fuzzy Topological Spaces and FRi-Proximities", Fuzzy Sets and Systems , 27 , 217-231 , (1988).

[6] Kandil , A. , Saleh, S. and Yakout, M. M. , " Fuzzy Topology On Fuzzy Sets : Regularity And Separation Axioms", American Academic and Scholarly Research Journal , No.2 , Vol. 4 , (2012).

[7] Kider , J. R. "Fuzzy Locally Convex-Algebras"Ph . D. Thesis , School of Applied Sciences , Univ. Technology ,(2004).

[8] Mahmuod, F.S., FathAlla M.A, and AbdEllah S.M," Fuzzy topology on Fuzzy sets: Fuzzy semi continuity And Fuzzy separation axioms", APP M m 153 127-140),(2003).

[9] Mashhour A.S. and GhanimM.H."Fuzzy closure spaces"J.Math.Anal.And Appl.106,pp. 145-170(1985).

[10] P. M. Pu and Y. M. Liu "Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergence" Journal of Mathematical Analysis and Applications, vol. 76, no. 2, pp. 571-599(1980).

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**Abstract:** In this paper, we present the space of functions of bounded - variation in the sense of Riesz – Korenblum, denoted by [ ], which is a combinations of the notions of bounded – variation in the sense of Riesz and of bounded – variation in the sense of Korenblum. In the light of this, we prove that the space generated by this class of functions is Banach algebra with respect to a given norm and we give a brief characterization of the composition (Nemystkii) operator on the space [ ].

**Key words:** Banach algebra norm, Bounded k - Variation, composition (Nemystkii) operator.

[1] B. Korenblum (1975) "An extension of Nevanlinna theory", Acta mathematica, Vol. 135, No. 3-4, pp. 187-219.

[2] C. Rembling, Banach algebras. Academic Press: San Diego, 2010.

[3] D. Cyphert, J. A. Kelingos, (1985) "The decomposition of functions of bounded k-variation into difference of k-decreasing functions", Studia mathematica, Vol. 81, No. 2, pp. 185-195.

[4] D. Waterman (1972) On convergence of Fourier series of functions of Generalized Bounded Variation. Studia Maths. 44: 107-117.

[5] F. Riesz (1953) "Unter suchugen uber systeme intergrierbarer funktionen", mathematische Annalen, Vol. 69, No. 4, pp. 115-118.

[6] H. G. Dales, P. Aiena, J. Eschmeier, K. Laursen & G. A. Willis (2003). "Introduction to Banach algebra, Operators, and Harmonic Analysis", Cambridge University Press: United Kingdom.

[7] J. Banas (2010). Functions of two variables with bounded variation in the sense of Riesz, Journal of Mathematics and Applications, No 32, pp 5-23 (2010).

[8] J. Park "On the Functions of bounded – Variation(I)" Journal of Applied Mathematics and informatics, vol. 23, pp 487-498, 2010.

[9] M. Castillo, S. Rivas, M. Sonaja, I. Zea, (2013) "functions of bounded in the sense of Riesz-Korenblum, Hindawi publishing corporation, Journal of function spaces and applications, Vol. 2013, Article 718507.

[10] M. Adispavic (1986). Concept of Generalized Bounded Variation and the Theory of Fourier series, Internat. J. Math. & Sci. Vol. 9 No. 2 : 223-244.

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Paper Type |
: | Research Paper |

Title |
: | KummerDirichlet Distributions of Matrix Variate in the Complex Case |

Country |
: | India |

Authors |
: | Ms. Samta Gulia, Prof. (Dr.) Harish Singh |

: | 10.9790/5728-0941321 |

**Abstract:**The aim of this paper is to investigate matrix variate generalizations of multivariate Kummer-Beta and multivariate Kummer-Gamma families of distributions in the complex case. The multivatiateKummer-Beta
and multivariate Kummer-Gamma families of distributions have been proposed and studied recently by Ng and
Kotz. These distributions are extensions of Kummer-Beta and Kummer-Gamma distributions. Many known or
new results have been made with the help of multivatiateKummer-Beta and multivariate Kummer-Gamma
families of distributions.

[1] C. Armero and M. J.Bayarri, A Bayesian analysis of a queneing system with unlimited service, J. Statist. Plann. Inference 58

(1997), no. 2, 241-261. CMP 1 450015.Zbl 880.62025.

[2] M.Gordy, Computationally convenient distributional assumptions for common-value auctions, Comput. Econom. 12 (1998), 61-78.

Zbl 912.90093.

[3] A. K. Gupta and D. K. Nagar, Matrix Variate Distributions, Chapman & Hall/ CRC Monographs and Surveys in Pure and Applied

Mathematics, vol. 104, Chapaman& Hall/CRC, Florida, 2000. MR 2001d:62055. Zbl 935.62064.

[4] W.R. Javier and A.K. Gupta, On generalized matric variate beta disgtributions, Statistics 16 (1985), no. 4, 549-557, MR

86m:62023. Zbl 602.62039.

[5] D.K. Nagar and L. Cardeno, Matrix variateKummer – Gamma distribution, Random Oper. Stochastic Equations 9 (2001), no. 3,

207-218.

[6] D.K. Nagar and A.K. Gupta, Matrix variateKummer-Beta distribution, to appear in J. Austral Math.Sec.

[7] K. W. Ng and S. Kotz, Kumer-Gamma and Kummer-Beta univariate and multivariate distributions, Research report, Department of

Statics, The University of Hong Kong, Hong Kong, 1995.

[8] Arjun K. Gupta: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403-0221,

USA.

[9] Liliamcardeno and Daya K. Nagar: Departmento de Mathematicas, Universidad de Antioquia, Medellin, A. A. 1226, Colombia.

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**Abstract:**In Nigeria there is no recognized scientific method of discriminating and classifying babies statistically into groups of study.

The purpose of this study includes to set up a discriminant function and classification rule that can be used to classify babies into two groups; to estimate the proportion of observations in each of the prior group; and to estimate the probability of correct classification and misclassification respectively. To this effect, a sample of 270 cases (infants) was observed with the following measurements: Age of mother (x1), weight at 36th week (x2), birth weight (x3), Parity (x4), Gestation Period (x5), and sex of the baby (x6). The birth weight was used to do the initial classification. Group 1 termed underweight (< 2.5kg) and Group 2 termed normal weight (≥ 2.5kg). We observed that the Discriminant Function Z= -0.02947228X1- 0.0514773X2- 8.130338X3 + 0.062259X4 + 0.0946538X5 + 0.5888918X6. Also 95.8 % of the original grouped cases were correctly classified. The percentage of misclassification is 4.15%. Conclusively the measure of the predictive ability which is the percentage of correct classification shows that discriminant analysis can be used to predict infants into two classes of weight and can also be used to predict group membership of any subject matter.

**Keywords:** Dicriminant, Classification, Multivariate, Misclassification, Gestational age, Confusion Matrix

[1] Adimora, G.N, Nigerian Journal of Clinical Practices. Vol 7.2004, pg 33- 36 Official Publication of the Medical and Dental

Consultants Association of Nigeria.

[2] Deswal B. S., Singh J. V., Kumar D.- A Study of Risk Factors for Low Birth Weight, Indian J. Community Med. 24, 2008: 127-

131

[3] Philip, J. Disala, Clinical Gynaecologicon.cology.(4th Edition). Mosby year book Inc. 1995

[4] Geoffrey .V.P Chamberline, Gynaecology by Ten Teachers.(16th Edition). ELBS. 1988.

[6] Anderson, T.W . An Introduction toMultivariate Statistical Analysis. New York: Wiley. 1984.

[7] Barnett,V. (ed.), Interpreting Multivariate Data. Wiley. 1981

[8] David,W.Stockburger, Multivariate Statistics: Concepts, Models and Applications. 1998

[9] Nwobi and Nduka . Statistical Notes and Tables for Research, Second Edition. Alphabet Nigeria Publishers. 2003

[10] Nwachukwu V.O, Principles of Statistical Inference. Second Edition. Zelon Enterprises. 2006.

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**Abstract:**In a linear motion of a system of two satellites connected by extensible cable, one stable equilibrium point exists when perturbative forces like Solar radiation pressure, shadow of the earth, oblateness of the earth, air resistance and earth's magnetic force act simultaneously. We have obtained one stable point of equilibrium
in case of perturbative forces like atmospheric resistance, Magnetic Force and oblateness of the earth acting
together on the motion of a system of two satellites connected by extensible cable in the central gravitational
field of earth in case of circular orbit of the centre of mass. We have used Liapunov's theorem on stability to examine the stability of the equilibrium point.

**Keywords:** Stability, Equiliribium Point, circular orbit, Liapunov Theorem, Satellites, perturbative forces.

[1]. Beletsky, V. V.: About the Relative Motion of Two Connected Bodies in orbit. Kosmicheskiye Issoledovania, vol. 7, No. 6, pp. 827 - 840, 1969 (Russian).

[2]. Thakur ; H.K. : The motion of a system of two satellites connected by extensible cable ; Ph.D. thesis, submitted to B. R. A. Bihar
University, Muzaffarpur, 1975

[3]. Singh, R. B.: Three Dimensional motion of system a two cable-connected satellite in orbit. Astronautica, acta, vol. 18, pp. 301 -
308, 1973

[4]. Singh, A. K. P. : Effect of Earth's Shadow on the motion of a system of two Satellites connected by extensible cable under
the influence of solar radiation pressure, Ph. D. Thesis Submitted to B. R. A. Bihar University, Muzaffarpur, 1990.

[5]. Kumar. V and Kamari. N, Stability of the equilibrium point of the Centre of mass of an extensible cable-connected satellites system in case of circular orbit in three dimensional motions, IJSER Volume 4, Issue9, September 2013 Edition (ISSN 2229-5518).

[6] Kumar. V and Kumari. N, Effect of the earth's oblateness, the shadow of the earth due to the solar radiation pressure and magnetic
force on the motion and stability of two satellites connected by an extensible cable in circular orbit of the centre of mass. IOSR-JM
Volume 10, Issue 10, Oct. 2013

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**Abstract:** In the present paper is an investigation of steady MHD free convection, heat and mass transfer flow of an incompressible electrically conducting fluid past an inclined stretching sheet under the influence of an applied uniform magnetic field with Hall current and radiation effect. Using suitable similarity transformations the governing fundamental boundary layer equations are transformed to a system of non-linear similar ordinary differential equations for momentum, thermal energy and concentration equations which are then solved numerically by the shooting method along with Runge- Kutta fourth-fifth order integration scheme. The results presented graphically illustrate that primary velocity field decrease due to increase of magnetic parameter, Angle of inclination, Dufour number, Prandtl number, Heat generation and Soret number while secondary velocity also decrease for Hall parameter . Other parameters increase the velocities of the fluid flow. Temperature field increases in the presence of Dufour number, heat generation, Schmidt number, Magnetic parameter, Grashof number & Modified Grashof number and decreases for other parameters. Also, concentration profiles decreases for increasing the values of Dufour number, Schmidt number, Heat generation, Soret number, Grashof number & Modified Grashof number but concentration increases for other parameters. The numerical results concerned with the primary velocity, secondary velocity, temperature and concentration profiles effects of various parameters on the flow fields are investigated and presented graphically. Also the skin friction coefficient, the local Nusselt number and the local Sherwood number are presented in Tables 1-3.

**Keywords:** Hall current, Heat generation, MHD, Radiation effect, Stretching Sheet.

[1] E.M.A. Elbashbeshy, M.A.A. Bazid, Heat transfer over an unsteady stretching surface, Heat Mass Transfer 41 ,2004, 1–4.

[2] Saleh M. Alharbi1, Mohamed A. A. Bazid2, Mahmoud S. El Gendy, heat and mass transfer in MHD visco-elastic fluid flow

through a porous medium over a stretching sheet with chemical reaction, Applied Mathematics, 1, 2010, 446-455.

[3] M. A. Seddeek and M. S. Abdelmeguid, "Effects of Radiation and Thermal Diffusivity on Heat Transfer over a Stretching Surface

with Variable Heat Flux," Physics Letters A, Vol. 348, No. 3 6, January 2006, pp. 172-179.

[4] A. A. Afify, "Similarity Solution in MHD: Effects of Thermal Diffusion and Diffusion Thermo on Free Convective Heat and Mass

Transfer over a Stretching Surface Considering Suction or Injection," Communications in Nonlinear Science and Numerical

Simulation, Vol. 14, No. 5, May 2009, pp. 2202-2214.

[5] Akiyama M, and Chong,, Numerical heat transfer Part A:,32 (1997) 419-33.

[6] Sudha Mathew, P. Raveendra Nath and N. B. V. Rama Deva Prasad, : Hall Effects On Heat and Mass Transfer Through a Porous

Medium In a Rotating Channel With Radiation, Advances in Applied Science Research, 3 (5), 2012, 3004-3019.

[7] R. Kumar and K. D. Singh,: Mathematical modeling of soret and hall effects on oscillatory MHD free convective flow of radiating

fluid in a rotating vertical porous channel filled with porous medium, Int. J. of Appl. Math and Mech. 8 (6), 2012, 49-68.

[8] L.J. Grubka, K.M. Bobba, Heat transfer characteristics of a continuous stretching surface with variable temperature, ASME J. Heat

Transfer 107, 1985, 248–250.

[9] C.H. Chen, Laminar mixed convection adjacent to vertical, continuously stretching sheets, Heat Mass Transfer 33, 1998, 471–476.

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**Abstract:** The paper examines Hausa riddles and games pertinent for the development of Mathematical thinking as a reconstruction focus. The paper reviewed literature relevant to the topic and the writer interviewed students, lecturers and other members of the society in kano to gather some examples of Hausa riddles and games. The findings revealed that there are so many numbers of Hausa riddles and games very relevant to the development of mathematical thinking which mathematics teachers and students could use to improve mathematics teaching and learning at the same time can be used for recreational activities and psychological tension release.

[1]. Bansilat, James &Naidoo, M. (2010): Whose voice matters? Learners South African Journal of Education. 30 (153-165).

[2]. Bode, B.H. (1933): The Confusion in Present Day Education in W.H. Kilpatrick (Ed): The Educational Frontier, Pp. 3-31. New York: Appleton Century.

[3]. Counts, G.S (1932): Dare the School Build a New Social Order?

[4]. Cremin, L.A. (1977): Tradition of American Education. New York Basic.

[5]. David, K. (1988): The Art of Reasoning. New York. W. W Norton and Company.

[6]. Putnam, R.T &Borko, H. (1997): Teacher Learning Implications of new Views of Cognition in B.J. Biddle, T.L. Good and L.F Godson (eds). International Handbook of Teachers and Teaching.Netherland. Kluwer Academic Publishers, 1223-1296.

[7]. Eshiwani (1979): In AMUCHMA-NEWSTELLER- 9 august, 1991.

[8]. Gardes, P. (2985): In AMUCHMA-NEWSTELLER- 9 august, 1991.

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Paper Type |
: | Research Paper |

Title |
: | The Value Distribution of Some Differential Polynomials |

Country |
: | India |

Authors |
: | Subhas S. Bhoosnurmath, K.S.L.N.Prasad |

: | 10.9790/5728-0945256 |

**Abstract:**We prove a value distribution theorem for meromorphic functions having few poles from which we
obtain several interesting results which improve some results of W. Doeringer, C.C.Yang, A.P.Singh,
G.P.Barker and others.

[1] BARKER G. P. and SINGH A. P. (1980): Commentarii mathematici universitatis : Sancti Pauli 29, 183.

[2] GUNTER FRANK and SIMON HELLERSTEIN (1986) : "On the meromorphic solutions of non homogeneous linear differential equation with polynomial co-efficients‟, Proc. London Math. Soc. (3), 53, 407-428.

[3] HAYMAN W. K. (1964) : Meromorphic functions, Oxford Univ. Press, London.

[4] HONG-XUN YI (1990): "On a result of Singh‟, Bull. Austral. Math. Soc. Vol. 41 (1990) 417-420.

[5] HONG-XUN YI (1991): "On the value distribution of differential polynomials", Jl of Math. Analysis and applications 154, 318-328.

[6] R. NEVANLINNA (1970) : Analytic functions, Springer-Verlag, New York.

[7] SINGH A. P. and DUKANE S. V. (1989): Some notes on differential polynomial proc. Nat. Acad. Sci. India 59 (A), II.

[8] SINGH A. P. and RAJSHREE DHAR (1993): Bull. Cal. Math. Soc. 85, 171-176.

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Paper Type |
: | Research Paper |

Title |
: | An Application of Random Fixed Point Theorem in Integral Equation |

Country |
: | India |

Authors |
: | Mukti Gangopadhyay, Pritha Dan, M. Saha |

: | 10.9790/5728-0945761 |

**Abstract:** In this paper random fixed point theorem has been applied seeking solution of Volterra type integral
equation involving more generous kernel.
2000 Mathematics Subject Classifications: Primary 47H10; Secondary 54H25, 60H25

**Key words and Phrases:** random fixed point, Caratheodory function, random Volterra integral equation

[1]. A. Spacek, Zufallige Gleichungen, Czechoslovak Mathematical Journal, 5(80), 1955, 462-466.

[2]. A. C. H. Lee and W.J.Padgett, Random Contractors with random nonlinear majorant functions, Nonlinear Analysis, TMA, 3, 1979,

707-715.

[3]. A. T. Bharucha-Reid, Random Integral Equations, Acad. Press New York, 1972.

[4]. C. P. Tsokos, On a stochastic integral equation of the volterra type, Math. Systems Theory, 3, 1969, 222-231.

[5]. Ding Xieping and Wang Fan, Solutions for a system of nonlinear random integral and differential equation under weak topology,

Applied Mathematics and Mechanics, Su, Shanghai, China, 18(8), 1997,721-737.

[6]. Ding Xieping, Criterion for the existence of solutions to random integral and differential equations, Applied Mathematics and

Mechanics,6(3),1985,269-275

[7]. H.W. Engl, A general stochastic fixed point theorem for continuous random operators on stochastic domaine, J. Math. Anal. Appl.,

66(1),1978, 220-231.

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**Abstract:** The present paper deals with the determination of displacement and thermal transient stresses in a thick ( ) circular plate with internal heat generation. External arbitrary heat supply is applied at the upper surface of a thick ( ) circular plate, whereas the lower surface of a thick ( ) circular plate is insulated and the heat is dissipated due to convection in surrounding through lateral surface. Here we compute the effect of Michell function on the thickness of circular plate with internal heat generation. The governing heat conduction equation has been solved by using integral transform method and the results are obtained in series form in terms of Bessel's functions and the results for temperature change and stresses have been computed numerically and illustrated graphically.

**Keywords:**Thick plate ( ) Thin plate ( ), internal heat generation, thermal stresses.

[1]. Nowacki, The state of stresses in a thick circular disk due to temperature field, Bull. Acad. Polon. Sci., Ser. Sci. Techn., 5, 227, (1957) .

[2]. Roy Choudhary S. K., A note of quasi static stress in a thin circular plate due to transient temperature applied alongthe circumference of a circle over the upper face, Bull Acad. PolonSci, Ser, Scl, Tech., 20-21, (1972).

[3]. J. N. Sharma, P. K. Sharma and R. L. Sharma, Behavior of thermoelastic thick plate under lateral loads, Journal of Thermal Stresses, 27, 171-191, (2004).

[4]. V. S. Gogulwar and K. C. Deshmukh, Thermal stresses in a thin circular plate with heat sources, Journal of IndianAcademy of Mathematics, 27, (1), (2005). [5]. V. S. Kulkarni and K. C. Deshmukh, Quasi-static transient thermal stresses in thick circular plate,Journal of Brazilian Society of Mechanical Sciences and Engineering,, 30, no.2, 172-177, (2008).

[6] Bhongade C. M. and Durge M. H., Effect of Michell function on steady state behavior of thick circular plate, IOSR Journal of Mathematics,8(2), 55-60,( 2013).