iosr-Jm   Volume-1 ~ Issue-5

Abstract: Quality human resources enhance the progress and prosperity of any nation. Excellent educational system produces good citizens. Our current educational system should be revitalized which produces creative, talented and co-operative people according to the recent global pressure. In the present 21st century the explosion of technologies uplift the world into the sky. It leads to globalization. We require powerful brainy citizens for this competitive world. Education can give tremendous boost to these citizens in the global society. Education should not only reflect the needs of the society but also excellence. Every effort should make to adopt our educational system today's changing economic and social realities of the scientific world. No branch of science is complete without mathematics. Mathematical understanding and reasoning are essential components of success in all walks of life. How did this precious mathematics subject teach or learn? Being abstract nature of mathematics most of the student find difficult to perceive it. Teachers are constantly looking for ways or tools to help their pupils understand the underlying concepts of the lessons. Attention is too paid on the integrative efforts of Information Processing Approach, Transformation between Short Term Memory and Long Term Memory and accelerating cognitive strategies. In this situation the investigators adopt some remedial measures through Multisensory strategies to overcome from these difficulties and enhance the understanding of the learners. This paper focuses about it.

[1]. Dr.Swaruparani ―Teaching of mathematics‖, 2007, APH publication, NewDelhi.
[2]. Mrs.C.Mattuvarkuzhali―Teaching of mathematics, 2009, APH publication, NewDelhi.
[3]. www.learning disabilities.com
[4]. www.dylexia for teacher.com
[5]. www.housing.sc.edu

Abstract:Our aim of this paper is to find a Eulerian Integral and a main theorem based on the fractional operator associated with generalized polynomial and a multivariable I-function having general arguments. The theorem provides extension of various results. Some special cases are also given.
Keywords-Fractionalintegral,Eulerian integral,multivariable I-function,Riemann-Liouville operator,Lauricella function.

[1]. Y.N. Prasad , Multivariable I-Function, Vijnana Parishad Anusandhan Patrika 29(1986) 231 – 235.
[2]. H. M. Shrivastava and M A Hussain, Fractional integration of the H-Function of several variable, compute, Math, April 30(1995),73-85.
[3]. V.B.L Chaurasia and V.K Singhal,Fractional integration of certain special functions,Tamkang J.Math.35(2004),13-22.
[4]. A.P.Prudnikov,Yu.A. Brychkov and O.I.Marichev,Integrals and series,Vol.I,Elementary Functions,Gordon and Breach,Newyork-London-Paris-Montreux-Tokyo,1986.
[5]. M. Saigo and R.K.Saxena ,Unified fractional integral formula for the multivariable H-function,J.Fract.Calc.15 (19999),91-107.
[6]. R.K. Saxena and K.Nishimoto,Fractional integral formula for the H-function,J.Fract.Calc. 13 (1994),65-74.
[7]. R.K. Saxena and M. Saigo, Fractional integral formula for the H-function II,J.Fract.Calc. 6 (1994),37-41.
[8]. H.M.Srivastava and M.C.Daoust,Certain generalized Neumann expansions associated with the Kampe de Feriet function,Nederl.Acad.We-tench.Indag.Math. 31 (1969),449-457.
[9]. H.M. Srivastava , K.C. Gupta and S.P. Goyal,The H-functions of One and Two Variables with Applications,South Asian Publishers,New Delhi-Madras,1982.

Abstract:In this paper we study the unseady Convective Heat Transfer flow of a viscous electrically conducting
fluid in a vertical wavy Channel under the influence of an inclined magnetic field. The unsteadiness in the flow
is due to an Oscillatory flux in the fluid region. The equations governing the flow and Heat Transfer which are
Non-linear coupled in nature are solved by employing a perturbation technique with the slope  of the wavy
walls as perturbation parameter the influence of Hall effects the radiation and Heat sources on the flow and
Heat Transfer characteristics has been studied graphically the average Nusselt Number on the boundary walls
  1 are numerically evaluated for different values of ,β,  and N.

[1]. Christopher Philip, G, Heat and Mass transfer from a film into steady shear flow, J. I. Mech. Applied Matha, v. 43 (1990).
[2]. Lai F.C : Coupled heat and mass transfer by natural convection from a horizontal line source in saturated porous medium. Int. Comm. Heat Mass transfer, v. 17 pp, 489-499 (1990).
[3]. Chen, T.S, Yuh, C.F and Montsoglo, H : Combined Heat and Mass transfer in mixed convection along vertical and inclined planes. Int. J. Heat Mass transfer, v. 23, pp, (527-537) (1980).
[4]. Poulikakos, D. : On Buoyancy induced heat and mass transfer from a concentrated surface in a infinite porous medium, Int. J. Heat Mass transfer, v. 28, No. 3, pp, 621-629 (1985).
[5]. Angirasa, D, Peterson, G. P, Pop I : Combined heat and mass transfer by natural convection with buyancy effects in a fluid saturated porous medium, Int. J. Heat mass transfer v. 40, no. 12, pp, 2755-2773 (1997)
[6]. B. Hossain, M.D.A and H.S. Thakar: Radiation effects on mixed convection along a vertical plate with uniform surface temperature heat and mass transfer, V.34, PP. 243-248, (1996).
[7]. Ching-Yang Cheng : Natural convection Heat and Mass transfer near a vertical wavy surface with constant wall temperature and concentration in a porous medium, Int. Comm. Heat Mass transfer v. 27, No. 8, pp, 1143-1154 (2000).
[8]. Cess, R.D. The interaction of thermal radiation with free convection of heat transfer, Int. J. Heat Mass transfer v. 9, pp, 1269-1277 (1966).

Abstract: In this paper, we introduce and investigate a new class of set called ω - λ -open set which is weaker than both ω -open and λ -open set. Moreover, we obtain the characterization of λ -Lindelof spaces.
Keywords: Topological spaces, -open sets, λ -Lindelof spaces.
2000 Mathematics Subject Classification: 54C05, 54C08, 54C10.

[1] F.G.Arenas, J.Dontchev and M.Ganster, On -closed sets and dual of generalized continuity, Q&A Gen.Topology, 15, (1997), 3-13.
[2] M.Caldas, S.Jafari and G.Navalagi, More on -closed sets in topological spaces, Revista Columbiana de Matematica, 41(2), (2007), 355-369.
[3] H.Maki, Generalized  -sets and the associated closure operator, The special issue in commemoration of Prof. Kazusada IKEDA's Retirement, (1.Oct, 1986), 139-146.

Abstract: We have critically described the various techniques for proving termination of term rewriting systems and have shown that the best of these methods is the transformation method. Transformation method implies that the termination of a given term rewriting system can be concluded from the termination of the transformed one, and proving termination of the transformed term rewriting system is often easier than proving termination of the given term rewriting system directly. The transformation method may be applied to prove termination of a term rewriting system where standard methods fail. 1991 Mathematics Subject Classification: 68Q42, 68-02
Keywords: Contractum, Non-erasing, Rewriting, Terms, Termination.

[1] N. Dershowitz, Termination of Rewriting, Journal of Symbolic Computation, 3(1 &2), 1987, 69-115.
[2] Terese, Term Rewriting Systems, in M. Bezem, J. W. Klop and R. Vrijer (Eds.), Cambridge Tracts in Theoretical Computer Science, 55 (Cambridge University Press, 2003).
[3] J. W. Klop, Term Rewriting Systems, in S. Abramsky, D. Gabbay, and T. Maibaum,(Eds.), Handbook of Logic in Computer Science, 1( Oxford University Press, 1992) 1-112.
[4] G. Huet and D. S. Lankford, On the uniform halting problem for term rewriting systems. Rapport loboria 283, Institut de Recherche en Informatique et en Automatique, Le Chesnay, France, 1978.
[5] F. Baader and T. Nipkow, Term Rewriting and All That (Cambridge University Press, 1998).
[6] M. Dauchet, S. Tison, T. Heuillard, and P. Lescanne, Decidability of the confluence of ground term rewriting systems, Proceedings of the 2nd Symposium on Logic in Computer Science, New York, USA, 1987, 353-359.
[7] M. Oyamaguchi, The Church-Rosser property for ground term rewriting systems is decidable. Theoretical Computer Science 49 (1),1987.
[8] H. P. Barendregt, The Lambda Calculus, its syntax and semantics, Studies in Logic and the Foundations of Mathematics 103, 2nd edition, Netherlands, 1984.
[9] S. Blom, Term graph rewriting: syntax and semantics, IPA dissertation series, no. 2001-05, 2001.
[10] C. March𝑒 and H. Zantema, The Termination Competition,. in F. Baader (Ed.), Proceedings of the 18th International Conference on Rewriting Techniques and Applications, LNCS 4533, Springer Verlag, 2007, 303-313.
[11] N. Dershowitz, Ordering for Term Rewriting Systems, J. Theoretical Computer Science, 17(3), 1982, 279-301.
[12] A. Koprowski, Certification of Termination Proofs for Term Rewriting, Radboud University Nijmegen, Foundations Group, intelligent Systems, ICIS, 2008.
[13] D. S. Lankford, Canonical Algebraic Simplification in Computational Logic, Memo ATP-25, Automatic Theorem Proving Project, University of Texas, USA, 1975.

Abstract: This paper proposed a new generalization of Sam-Solai's Multivariate Wigner distribution of Kind-1 of Type-A from the univariate case. Further, we find its Cumulation, Marginal, Conditional distributions, Generating functions and also discussed its special case. The special cases include the transformation of Samsolai's Multivariate Wigner distribution of Kind-1 of Type-A into Multivariate one parameter Wigner distribution of Kind-1 of Type-A, Multivariate Wigner distribution of Kind-1 of Type-B, Multivariate log-Wigner distribution of Kind-1 of Type-A and Multivariate Inverse -Wigner distribution of Kind-1 of Type-A. It is found that the conditional variance of Sam-Solai's Multivariate conditional Wigner distribution is heteroscedastic and the correlation was found to be -0.16. Area values of the bi-variate Wigner surface also extracted and Wigner surfaces, contours are also visualized.
Keywords: Sam-Solai's Multivariate Wigner distribution of Kind-1 of Type-A, Multivariate one parameter Wigner distribution of Kind-1 of Type-A, Multivariate Wigner distribution of Kind-1 of Type-B, Multivariate log-Wigner distribution of Kind-1 of Type-A and Multivariate Inverse -Wigner distribution of Kind-1 of Type-A

[1] E. Wigner. "On the Distribution of the Roots of Certain Symmetric Matrices." Ann. of Math., 67, 1957, 325-327
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[3] F.A. Berezin Some remarks on the Wigner distribution, Teoret. Mat. Fiz., 17,1973, 305–318
[4] Mardia,"The von Mises Distribution Function," Applied Statistics, 24, 1975, (pp. 268–272)
[5] Geoffrey S. Watson, Distributions on the Circle and Sphere, Journal of Applied Probability Vol. 19, Essays in Statistical Science 1982 pp. 265-280
[6] Watson, G.S. , Statistics on Spheres, New York: Wiley, 1983.
[7] Z. D. Bai, Y. Q. Yin. Convergence to the Semicircle Law, The Annals of Probability.Volume 16, Number 2 , 1988, 863-875
[8] Fang, K.T., Kotz, S., and Ng, K.W. , Symmetric Multivariate and Related Distributions, London: Chapman and Hall,1990.
[9] Gradshteyn, I.S., and Ryzhik, I.M. , Table of Integrals, Series, and Products, fifth edition edited by A. Jeffrey, San Diego: Academic Press, 1994
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[11] A. Boutet de Monvel and A. Khorunzhy On the Norm and Eigen value Distribution of Large Random Matrices, Ann. Probability, 27,1999, 913–944
[12] Evans, M., Hastings, N., and Peacock, B "von Mises Distribution." Ch. 41 in Statistical Distributions, 3rd ed. New York. Wiley, 2000.
[13] Hiai, F. and Petz, D The Semicircle Law, Free Random Variables and Entropy. Mathematical Surveys and Monographs 77, American Mathematical Society, Providence,2000.
[14] Shimizu.K., and Iida, K. ,"Pearson Type VII Distributions on Spheres," Communications in Statistics – Theory and Methods, 31, 2002, 513–526

Abstract: Roter (1964) and Matsumoto (1969) have studied on second order recurrent spaces and Kaehlerian spaces with parallel or vanishing Bochner curvature tensor. Singh (1972) have defined and studied Kaehlerian recurrent and Ricci- recurrent spaces of second order. Further, Rawat and Prasad (2010) studied on holomorphically projectively flat parabolically Kaehlerian spaces. In the present paper, we have studied Einstein – Kaehlerian recurrent and symmetric spaces of second order and several theorems have been derived within. The necessary and sufficient condition for an Einstein – Kaehlerian conharmonic recurrent space to be Kaehlerian recurrent space is discussed.
Key words: Recurrent, symmetric, Kaehlerian spaces.

[1] Matsumoto, M. : On Kaehlerian spaces with parallel or vanishing Bochner curvature tensor, Tensor, N. S. , 20(1), 25- 28 (1969).
[2] Negi, D. S. and Rawat, K. S. : Some bi- recurrence and bi- symmetric properties in a Kaehlerian space, Acta Cien. Ind., Vol. XX,M, No.1, 95-100 (1994).
[3] Rawat, K. S. and Gyan Prakash : Some recurrence and symmetric properties of a Kaehlerian space, Acta Ciencia Indica , Vol. XXX M, No.4, 701-704 (2004).
[4] Rawat , K. S. and Girish Dobhal : On the bi- recurrent Bochner curvature tensor, Jour. of the tensor society , Vol.1 , 33-40 (2007).
[5] Rawat, K. S. and Kunwar Singh : Some bi- recurrence properties in a Kaehlerian space, Jour. PAS , Vol. 14 (Mathematical Science) pp. 199-205 (2008).
[6] Rawat, K. S. and Girish Dobhal : Study of the decomposition of recurrent curvature tensor fields in a Kaehlerian recurrent space, Jour. Pure and Applied Mathematica Sciences, Vol. LXVIII , No. 1-2 , Sept (2008).
[7] Rawat, K. S. and Virendra Prasad : Some recurrent and symmetric properties in an almost Kaehlerian space , Jour. PAS , Vol. 14 (Mathematical Sciences), pp. 283-288 (2008).
[8] Rawat, K. S. and G. P. Silswal: Theory of Lie-derivatives and motions in Tachibana spaces, News Bull. Cal. Math. Soc. , 32, (1-3), 15-20 (2009).
[9] Rawat K. S. and Mukesh Kumar: On curvature collineations in a Tachibana recurrent space, Aligarh Bull. Math., 28 No. 1-2, 63-69 (2009) MR 2769016.
[10] Rawat, K. S. and Virendra Prasad: On holomorphically projectively flat parabolically Kaehlerian spaces, Rev. Bull. Cal. Math. Soc. , 18, (1), 21-26 (2010).

Abstract: A given theorem is a some advanced proof in product summability of infinite series.Many other results some known and unknown are derived .
Key Words : And Phrases: summability, absolute summability, product summability.

[1] DAS, G.-Tauberian theorem for absolute Nӧrlund summability, 'Procidings of the London Math. Soc. Vol. 19, No. 2, pp 357-384, 1969'.
[2] FLETT, T.M. - On an extension of absolute summability andsome theorems of Littlewood and Paley, Prociding of London Math. Soc. Vol. 7 , No.1, pp.-113-141, 1957.
[3] SULAIMAN, W.T. – Note on product summability of infinite series, International J. of Math. And Math. Sci. Vol. ,2008, Article I.D. 372604.