iosr-Jm   Volume-9 ~ Issue-3

Abstract: A numerical method based on parametric spline with adaptive parameter is given for the secondorder singularly perturbed two-point boundary value problems of the form 0 1  y  p(x)y  q(x)y  r(x); y(a)  ; y(b)  The derived method is second-order and fourth-order convergence depending on the choice of the two parameters  and  . Error analysis of a method is briefly discussed. The method is tested on an example and the results found to be in agreement and support the predicted theory. Keywords: Singular perturbation; parametric spline functions; BVPs; ODEs.

[1] A.K. Aziz, Numerical Solution of Two Point Boundary-Value Problem, Blaisdal, NewYork, 1975.
[2] G. Micula, S. Micula, Hand Book of Splines, Kluwer Academic Publishers, Dordrecht,London, Boston, 1999.
[3] H.G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly-Perturbed Differential Equations, Springer, New York, 1996.
[4] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, NewYork, 1962.
[5] P.M. Prenter, Spline E.A. As and Variational Methods, Wiley, New York, 1975.
[6] U.M. Ascher, R.M.M. Mattheij, R.D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice-Hall, Englewood Clis, NJ, 1988.

Abstract:In this paper we have tried to state that Kaprekar's constant is a fixed point of iterative function defined on set of all positive four digit numbers where none of its any three digits are equal. Hence consequently we have given the definition of Kaprekar's constant in terms of function.

Key words: Kaprekar's constant, Fixed point, Iterative function, Four digit numbers.

[1] Investigations into the Kaprekar Process by Robert W. Ellis and Jason R. Lewis
[2] An Interesting Property of the Number 6174, Scripta Math, 21(1955)304 by D R Kaprekar
[3] The Determination of Kaprekar Convergence and Loop Convergence of All Three-Digit numbers. American Monthly,95(1988,105-
112) by Klaus E. Eldridge and Seok Sangong
[4] The Kaprekar Routine and Other Digit Games for Undergraduate Exploration by K Peterson and H Pulapaka
[5] Mac Tutor History of Mathematics, Article by J J O‟Connor and E F Robertson

Abstract: Principal Component Analysis(PCA) is a data analysis tool that is used to reduce the dimensionality of a large number of interrelated variables while retaining as much of the information as possible. In this paper, PCA has been utilized on the crime data of Nigeria to discover the distinct influential variables; in addition which variables have silence in the identification of State being safe or dangerous. From the result, four Principal Components (PCs) have been retained using both scree plot and Kaiser's criterion which accounted for 75.024% of the total variation.

Key words: Principal Component Analysis (PCA), crime data.

[1] Cleen (2007). The Nigerian Police as at November 2007. http://www.cleen.org/sumarry%20crime% 20statistics% 20in%20nigeria%202007.p [2] Danbazau, A.B. (2007). Criminology Justice. 2nd edition. Ibadan: Spectrum Books Limited. [3] Ifeanyi (2004) Nigeria Crime Statistics [4] Tappan (1964:32) Definition of crime
[5] Wiki/cr (2009). Crime, http://en.wikipedia.org/wiki/criminal_94. [6] Lombroso, C. (1911). The Criminal Man, New York: Putman. [7] Sutherland, E.H. (1939). The white-colar criminal. American Sociological Review, 5: 1-12 [8] Oyebanji, J.O. (1982). Economic Development and the Geography of crime: an Empirical Analysis. [9] Akpan, A.U. (2002). Notion of causes of crime among Nigerians. International Journal of Social and Policy Issues: 1(1): 36-40 [10] Kutigi, I. (2008). Puberty, Ignorance and Evil Intentions as strong crime factors. Tribune, Monday 23/06/2008.

Abstract:In this paper, magneto hydrodynamics boundary layer heat transfer over a moving flat plate is discussed, using similarity transformation momentum and energy equations that are reduced in to nonlinear ordinary differential equations. The nonlinear differential equations are solved using implicit finite difference Keller box method. Graphical results of fluid velocity and temperature profile are presented and discussed for various parameters.

Key words: MHD boundary layer, moving surface, heat transfer, Keller box method.

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[3] Afzal N, Int. J. Heat Mass Trans. 15, 1972, 863.
[4] Sakiadis B C, Boundary layer behaviour on continuous moving solid surfaces: I. Boundary layer equations for two dimensional and axi-symmetric flow, II. Boundary layer on continuous flat surface, III. Boundary layer on a continuous cylindrical surface. AICHE Journal, 7, 1961, 26-28, 221-225, 467-472.
[5] Erickson L E, Fan L T and Fox V G, Heat and mass transfer on a moving continuous flat plate with suction or injection, Ind. Engg. Chem. Fundam.5, 1966, 19-25.
[6] Tsou F K, Sparrow E M and Goldstein R J, Flow and heat transfer in the boundary layer on a continuous moving surface, Int J Heat Mass Transfer, 10, 1967, 219- 235.
[7] R. N. Jat, Abhishek Neemawat,Dinesh Rajotia, MHD Boundary Layer Flow and Heat Transfer over a Continuously Moving Flat Plate, International Journal of Statistika and Mathematika,3(3),2012,102-108.
[8] Cebeci T., Bradshaw P. Physical and computational Aspects of Convective Heat Transfer, (New York: Springer, 1988), 391-411.
[9] Na T.Y., Computational Methods in Engineering Boundary Value Problem, (New York: Academic Press 1979), 111-118.