iosr-Jm   Volume-8 ~ Issue-6

Abstract: In this communication, we propose useful fuzzy entropy measure and study its particular cases. Some coding theorems have been proved for decipherable codes using Holder's inequalities. Some known results are the particular cases of our proposed measure. Keywords: Fuzzy set, Fuzzy entropy, Holder's inequalities, Decipherable codes. AMS subject Classification: 94A17, 94A24.

[1] J. Aczel, On Shannon's inequality, optimal coding and characterization of Shannon's and Renyi' entropies, Institute Nazionale di Alta Matematica, Symposia Mathematica XV (1975) 153-179.

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[8] Kapur, J. N., Measures of fuzzy Information, Mathematical Sciences Trust Society, New Delhi (1997).

[9] Klir,G.J.and B. Parviz, A note on the measure of discord, Proc. Eighth conference on Artificial Intelligence, San Mateo, California, (1992) pp.138-141.

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Abstract: In this paper, we prove that the Dumbbell graph, Star graph, Cartesian product n P C 2 and K1 + Cn are even-even graceful. The even-even graceful labeling of a graph G with q edges means that there is an injection f: E (G) → {2, 4,…, 2q} so that induced map f *: V (G) →{0, 2,…,(2k-2)} defined by f *(x) ≡ Σf(x, y) (mod 2k) where k = max {p, q} makes all distinct and even. Keywords: Even-even graceful labeling, Dumbbell graph, Star graph and wheel graph.

[1]. A.Rosa, "On certain valuations of the vertices of a graph", Theory of Graphs (Internat. Symposium, Rome, July 1966), Gordon and
Breach, N. Y. and Dunod Paris (1967) 349-355.
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Labelings of Trees of Diameter 5", AKCE J.Graphs. Combin., 6, No. 2 (2009) .
[5]. A. Solairaju and K. Chithra, Edge-odd graceful labeling of some graphs, Proceedings of the ICMCS, 1 (2008) 101-107.
[6]. R. Sridevi, S. Navaneethakrishnan, A. Nagarajan and K. Nagarajan," Odd-Even graceful graphs", J. Appl. Math. & Informatics Vol.
30(2012), No. 5 - 6, pp. 913 – 923.
[7]. J.A.Gallian, A dynamic survey of graph labeling, Electronic, J.Comb,
[8]. Sin-Min Lee,Kuo-Jue Chen,Yung-Chin Wang "On the Edge graceful spectra of cycles with one chord and dumbbell graphs",
Congressus Numerantium 170 (2004), pp. 171-183.

Abstract: In this article we introduce and study the notions of ( , ) s v r  -lacunary strongly summable, ( , ) s v r  - Cesàro strongly summable, ( , ) s v r  - statistically convergent and ( , ) s v r  -lacunary statistically convergent sequence of fuzzy numbers. Consequently we construct the spaces   ( , ) F s v r L  ,   1 ( , ) F s v r   ,   ( , ) F s v r S  and   ( , ) F s v r S  respectively and investigate the relationship among these spaces. Further we show that   ( , ) F s v r L  and   1 ( , ) F s v r   are complete metric spaces.

Keywords: Sequence of fuzzy numbers; Difference sequence; lacunary strongly summable; Cesàro strongly summable; statistically convergent; lacunary statistically convergent; Completeness. AMS Classification No: 40A05; 40D25.

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Abstract: A closed queuing network that consists of M nodes was studied, and in contrast to an open network, there is a constant population of K customers in the network with no external source or sink. Each node i is a first in first out queue, where the service time is drawn independently. That is, the service time is exponentially distributed with service rates i  i   n and a customer departing from queue i chooses j next with probability i j q , . The Gordon and Newel theorem with the arrival properties of a closed network were used to analyze the Mean values for the number of customers, sojourn time, waiting times and the algorithm for solving the mean value for closed networks were then derived with series of application considered.

[1]. Agboola, S.O. (2007): On the Analysis of M/G/1 Queues. Unpublished M.Sc. (Mathematics) Thesis submitted to Department of
Mathematics, University of Ibadan.
[2]. Agboola, S.O. (2011): The Analysis of Markov Inter-arrival Queues Model with K – Server under Various Service Points.
Unpublished M.Sc. (Statistics) Thesis submitted to Department of Mathematics, Obafemi Awolowo University, Ile-Ife.
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Abstract: Let M be a 2-torsion free prime Γ- ring and U a Lie ideal of M. Let F : M → M be a mapping defined by F (uαv) = F(u)αv + uk(α)v + uαd(v), for all u ,v  U and α  Γ. Then F is a generalized kderivation on U of M if there exists a k- derivation d on U of M. Also F is a Jordan generalized k- derivation on U of M if there exists a k - derivation d on U of M such that F(uαu) = F(u)αu + uk(α)u + uαd(u), for all u  U and α  Γ. In this article, we prove that every Jordan generalized k - derivation on a Lie ideal U of a 2 - torsion free prime Γ- ring M is a generalized k - derivation on U of M.

Keywords: Lie ideal, k- derivation, generalized k- derivation; Jordan generalized k- derivation, Prime Γ- ring.

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[7] M. M. Rahman and A. C. Paul, "Jodan generalized derivations on Lie ideals of prime Г- rings". South Asian Journal of Mathematics, 2013, vol 3(3): 148 - 153.
[8] A. C. Paul and Md . Sabur Uddin , Simple Gamma rings with involutions , IOSR Journal of Mathematics vol. 4, Issue 3 (Nov - Dec 2012 ) PP 40 - 48.
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[10] S. Chakraborty and A. C . Paul , k-Derivations and k- homomorphisms of Gamma rings , Lambert Academic Publishing CmbH Co, KG Heinrich - Beking - Str . 6 - 8 , 66121, saarbrchen, Germany, 2012.

Abstract: The present investigation consists of an analytic treatment of a peristaltic transport of a magneto non-Newtonian fluid through a porous medium in a finite channel under the assumptions of long wavelength and low Reynolds number. The fluid under consideration is obeying Eyring-Powell model. The velocity field, the pressure gradient, the local wall shear stress and the volume flow rate are obtained in the explicit form. Numerical results are graphically discussed for various values of physical parameters of interest. The results show that the pressure increases as both of the magnetic and the non-Newtonian parameters increasing. The range between peak and trough of local wall shear stress distribution increases with the increasing of the magnetic parameter while this range decreases as the non-Newtonian parameter increases. The reflux region decreases as the magnetic parameter increases but it increases as non-Newtonian parameter increases.

Key words: Eyring-Powell Fluid, Magnetic Field, Non-Newtonian, Porous Medium.

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Abstract: Squares of integers can be expressed as sum of consecutive odd numbers. Is there a general case for all powers? After a thorough search with available materials I could not find one such theorem. Here is an attempt in that lines. Sum of consecutive odd numbers as powers of integers and sum of consecutive even numbers also as powers of integers with formulae and simple proof.

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Abstract: Using two-phase sampling mechanism, a modified generalized exponential estimator of finite population mean in the presence of the available knowledge on the auxiliary variable x is suggested i.e. the population mean of auxiliary variable x is not known. The suggested estimator is found to be more efficient than usual two-phase ratio, product, regression estimators and also modified exponential ratio and product estimator suggested by Singh et.al (2007). Further a numerical study is made to support the theoretical comparison.

Keywords: Auxiliary variable, Exponential Estimator,Mean Square Error, Two-Phase Sampling,

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[ 6 ] Murthy , M.N. Sampling Theory and Methods, Statistical Publishing Society, Calcutta.(1977)

[ 7 ] Singh H.P. and Vishwakarma, G.K. Modified exponential ratio and product estimators for finite population mean in double sampling. Austrian journal of statistics, 36, No.3,(2007) 217-225.

[ 8 ] Steel R.G.D. and Torrie , J.H. . Principle and Procedures of Statistics, MC-Graw Hill, Newyork.(1960)

[ 9 ] Sukhatme, P.V.Sukhatme, B.V.Sukhatme, S. and Asok C. Sampling Theory of Surveys with Applications, Indian Society of Agricultural Statistics, New-Delhi.(1984)

Abstract: We present a mathematical model for the spread of dracunculiasis with focus on three populations; human, vector and parasite. The reproductive number is obtained from next generation matrix and the stability analyses of disease-free and endemic equilibria are conducted. Simulation of the model is presented by solving the systems of the differential equations to explore the behaviour of the model using maple 14. The paper also analyzes key parameters to determine the effective intervention. The result of this paper shows that reducing the parasite birth rate is more effective than water treatment.

Keywords: Copepod, dracunculiasis, reproductive number, stability.

[1] Anosike, J.C.; Azoro, VA; Nwoke, B.E.B; Keke, IR; Okere, AN; Oku, EE; Ogbul JN; Tony-Njoku, F: Okoro, O.U., Nwosu, D.c.Dracunculiasis in the North Eastern Border of Ebonyi State, South Eastern Nigeria. Int. J. Egg. Environm. Hith, 205, 2002, 1-7.
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Abstract: The main purpose of this paper to solve three dimensional thermoelastic problem of thermal stresses of semi infinite rectangular slab with internal heat source to determine temperature distribution, thermal displacements stress functions, at any point of the rectangular slab with given boundary conditions by applying the finite Marchi-Fasulo integral transform and infinite Fourier cosine transform technique.

Keywords: Rectangular slab, three dimensional thermoelastic problem, temperature distribution, thermal displacements, stress functions, integral transform.

[1] Khobragade,N.W., H.S.ROY, and S.H. Bagade: Thermal stresses of semi infinite rectangular beam. Internatinal journal of engineering and innovative technology(IJEIT), vol.3, july (2013).

[2] KHobragade N.W.and Parveen H.: Thermoelastic solution of a semi infinite rectangular beam due to heat generation. Journal of statistics and mathematics, Vol. (3), (2012).

[3] Tanigawa Y and Komatsubara Y.: Thermal stress analysis of a rectangular plate and its thermal stress intensity factor for compressive stress field, Journal of, Thermal Stresses, 20, 517. (1997)

[4] Adams, R.J. and Bert , C.W.: Thermoelastic vibrations of a laminated rectangular plate subjected to a thermal shock, Journal of Thermal Stresses, 22 875-895.(1999)

[5] Vihak, V.M., Yuzvyak, M.Y. and Yasinskij, A.V.: The solution of the plane thermoelasticity problem for a rectangular domain, Journal of Thermal Stresses, 21, 545-561. (1998)

[6] Khobragade, N.W. and Wankhede, P.C.: An inverse unsteady-state thermoelastic problem of a thin rectangular plate, The journal of Indian Academy of Mathematics, Vol. (25), No. 2, (2003).

Abstract: The steady two dimensional radioactive MHD boundary layer flow of an electrically viscous, incompressible, conducting fluid caused by a non-isothermal linearly stretching sheet placed at the bottom of fluid saturated porous medium in the presence of viscous dissipation, heat generation and chemical reaction is studied. The governing system of partial differential equations is converted to ordinary differential equations by using the similarity transformations and then solved by shooting method. The dimensionless temperature, velocity, and concentration are computed for different thermo physical parameters viz the magnetic parameter, permeability parameter, , wall temperature parameter radiation parameter, Prandtl number, heat generation parameter, Eckert number, Schmidt number and chemical reaction.

Keywords: Chemical reaction, Eckert number, Permeability parameter, Porous medium, Shooting method.

[1] M.A. Seddeek , and A.A. Almushigeh A.A , Effects of radiation and variable viscosity on MHD free convective flow and mass
transfer over a stretching sheet with chemical reaction, Appl. Appl Math, Vol. 5, No.1,2010, pp. 181-197.
[2] R.Kandasamy , T.Hayat , S.Obaidat S, Group theory transformation for Soret and Dufour effects on free convective heat and mass
transfer with thermophoresis and chemical reaction over a porous stretching surface in the presence of heat source/sink, Nucl. Eng.
Des., Vol. 241,No.6,2011, pp. 2155–2161.
[3] D.Pal , and B.Talukdar , Combined effects of Joule heating and chemical reaction on unsteady magnetohydrodynamic mixed
convection of a viscous dissipating fluid over a vertical plate in porous media with thermal radiation, Math. Comput. Modelling,
Vol. 54, No.11,2011,pp. 3016–3036.
[4] A.A. Joneidi , G.Domairry , and M. Babaelahi , Analytical treatment of MHD free convective flow and mass transfer over a
stretching sheet with chemical reaction, J. Taiwan Inst. Chem. Eng., Vol. 41No.1,,2010, pp. 35–43, 2010.
[5] P. Vyas , Radiative MHD flow over a non-isothermal stretching sheet in a porous medium, Applied Mathematical Sciences, Vol. 4,
No. 50,2010, pp. 2475-2484.

Abstract: The object of this paper is to obtain twenty five Eulerian type single integrals in the form of a general single integral involving multivariable extension of the Kampé de Fériet function [1]. The results are derived with the help of the generalized classical Watson's theorem obtained earlier by Lavoie et. Al. [2]. A few interesting special cases of our main result have also been discussed.

[1] Srivastava, H.M. and Panda, R.: Some analytic and asymptotic confluent expansions for functions of several variables. Math. Comput. 29 (1975), 1115-1128.

[2] Lavoie, J.L. et. Al.: Generalizations of Watson's theorem on the sum of a F23 , Indian J. Math. 32, (1992), no. 1, 23-32.

[3] Appell, P. et. Kampé de Fériet, J.: Fonctions Hypergeometriques et Hyper-spheriques Polynomes d'Hermite, Gauthier-Villars, Paris, (1926), p. 150 (Eq. 29).

[4] Karlsson, P.W.: Reduction of certain generalized Kampé de Fériet functions, Math. Scand. 32, (1973), 265-268.