iosr-Jm   Volume-15 ~ Issue-1 ~ Jan-Feb 2019

Abstract: In this paper, we proposed a three step method for approximating roots of non-linear equations. This method has three evaluations of function and first derivative which is modified from McDougall and Wotherspoon [1]. Numerical examples are tested to compare the method with other methods and demonstrate the efficiency of the proposed method.

Key Word: Non-linear equation, Iterative method, Newton's method, Multiple roots, Order of convergence

[1]. T.J. McDougall, and S.J. Wotherspoon, A simple modification of Newton's method to achieve convergence of order, Journal of
Applied and Mathematical letters, 29, 2014, 20-25.
[2]. A. Cordero, and J. R. Torregrosa, Low-complexity root-finding iteration functions with no derivatives of any order of convergence,
Journal of Computational and Applied Mathematics, 275, 2015, 502–515.
[3]. H. T. Kung, and J. F. Traub, Optimal order of one-point and multipoint iteration, Journal of the Association for Computing
Machinery, 21, 1974, 643–651.
[4]. F. Zafar, N. Yasmin, S. Akram, and M. D. Junjua, A general class of derivative free optimal root finding methods based on rational
interpolation, Science World Journal, 2015.

[5]. N.A. Mir, K. Bibi, and N. Rafiq, Three-step Method for Finding Multiple Root of Non-linear Equation, Life Science Journal, 11(7), 2014, 287-289.

Abstract: In this paper, some new types of separation axioms in topological spaces by using 𝛽g*-open sets are formulated. In particular the concept of 𝛽g*-R0 and 𝛽g*-R1 axioms are introduced. Several properties of these spaces are investigated using these axioms.

Key Word: 𝛽g*-open set, 𝛽g*-R0, 𝛽g*-R1, 𝛽g*-Ti(i= 0,1,2)

[1]. D.Andrijevic, semi preopen sets, Mat.Vesnik, 38(1) (1986), 24-32
[2]. K.Balachandran, P.Sundaram and H.Maki, On generalized continuous maps in topological spaces, Mem.Fac.sci.Kochi.Univ.Math.,12(1991),5-13.
[3]. C.Dhanapakyam ,K.ndirani,On 𝛽g*closed sets in topological spaces, Int. J. App. Research (2016),388
391
[4]. N.Levine, Generalized Closed sets in Topology, rend.Cir.Mat.palermo,2(1970),89-96. N.Levine, Semiopen sets and semi continuity in topological spaces.,Amer.Math.Monthly, 70(1963),36-41.
[5]. M.K.R.S Veerakumar, Between closed sets and g-closed sets, Mem. Fac. Sci.Kochi Univ.Ser.A,Math., 21 (2000) 1-19..

Abstract: This paper is devoted to define a new class of life distribution, named new better than used in increasing convex in Laplace transform order (NBUCL). A new test statistic for testing exponentiality against (NBUCL) class based on U-statistic is introduced. For the proposed test, the asymptotic properties are studied andselected critical values for sample size 5(5)50 are tabulated. The powers of this test are also estimated by using a simulation study for commonly used distributions in reliability. Pitman's asymptotic efficiencies of the test are calculated and compared with some old tests. The problem in the case of right censored data is also touched. Finally, our proposed test is applied to some real data sets in different areas.

Key Word: NBUCL class; Testing Exponentiality; U-statistic; Pitman asymptotic efficiency; censored data; Laplace transform.

[1]. Bryson MC, Siddiqui M. Some criteria for aging. Journal of the American Statistical Association. 1969;64:1472-83.
[2]. Marshall AW, Proschan F. Classes of distributions applicable in replacement with renewal theory implications. University of Rochester Rochester United States; 1972.
[3]. Cao J, Wang Y. The NBUC and NWUC classes of life distributions. Journal of Applied Probability. 1991;28:473-9.
[4]. Alzaid A, Kim JS, Proschan F. Laplace ordering and its applications. Journal of Applied Probability. 1991;28:116-30.
[5]. Denuit M. Laplace transform ordering of actuarial quantities. Insurance: Mathematics and Economics. 2001;29:83-102.