Abstract: This given note is a brief introduction to a new branch of Mathematics which comes under the category Exponentiation. Which deals with the study of base (of an exponential expression) and its split terms, explains the relation between the split terms, gives an extension for exponentiations (like Square, Cube etc.). That we can call as "Exponential Base Splitting Expressions-Extensions" simply abbreviated as "Exbaspex . The main objective of this branch is to provide a suitable extension for all exponential expressions and thereby increasing the possibilities and futher applications of exponentiations.

[1]. Wikipedia, the free encyclopedia. http://en.wikipedia.org/wiki

Abstract: In the present paper we have discuss numerical solution of seepage problem of ground water flow using hydraulic theory. We examine the seepage of groundwater down sloping bedrock in heterogeneous soil in vertical direction and numerical solution is obtained by RK-4 method with MatLab coding .
Keywords: Ground water, fluid flow, RK-4 method.

[1]. Crichlow, H.B.(1977). Modern Reservoir Engineering, A simulation approach; Printice-Hall, New York. Polubarinova-Kochina, P.Ya.(1962). Theory of Groundwater Movement Princeton Uni. Press. 416.
[2]. Verma,A.P.(1967). Seepage of groundwater in heterogeneous soil on sloping bedrock. Revue Roumanie des science Techniques, Mechanique Appliquce, Rumania, Tome 12 No.6,1185-1191.
[3]. Verma, A.P.(1965). Seepage of groundwater in two layered soil with an inclined boundary when lower layer is heterogeneous and upper one homogeneous. Journal of Science and Engineering Research, Kharagpur (India) Vol. IX. Part-I, 40-46.
[4]. Mehta M.N.(1977). Asymptotic expansion of flow through porous media,Ph.D. Thesis, South Gujarat Uni, Surat (India) 62-83.
[5]. Alyavuz, B.,Kocyigit, O.And Gultop, T.(2009). Numerical solution of seepage problem using Quad-tree based triangular finite elements. Int. J. of Engg. and Appl. Sci. (IJEAS). Vol.I, Issue 1(2009) 43-56.

Abstract: In this paper, an existence result for a nonlinear fourth order random differential equation is proved under Carathéodory condition. Two existence result for extremal random solutions are also proved for Carathéodory as well as discontinuous cases of nonlinearity involved in the equation. Our investigation are placed in the Banach space of continuous real valued function on closed and bounded interval of the real line together with an application of the random version of the Leray-Schauder Principle.

Keywords and phrases: Initial value problem, Random differential equation, Random fixed point theorem, Existence theorem, Extremal solution.

[1] A.T.Bharucha-Reid, Random Integral Equations Academic Press, New York London , 1972, volum96

[2] B.C.Dhage, S.V. badgire, S.K.Ntouyas, Periodic boundary valu problem of second order random differential equation .Electron.J.Qaul.Theory Diff.Equ.,21(2009), 1-14.

[3] D. Jianng, W. Gao, A. Wan A monoton method for constructing extremal to fourth order boundary value problem. Appl. Math. comput, 132 (2002)

[4] TihomirGyulov, Solvability of some nonlinear fourth order boundary value problem, Central European University.(2009)

[5] G.S. Laddhe, V.Lakshmikantham, Random Differential Inequalities, Academic Press, New York, 1980

[6] B.C.Dhage Some algebraic and topological random fixed point theorem with applications to nonlinear random integral equation .Tamkang J. Maths 35(2004), 321-345

[7] Xiaoling Han, Xuan Ma, Guowei Dai, Solution to fourth order random diff. equ. eith periodic boundary condition .,Electronic Journal of Diff. Equ. Vol.2012 no. 235, 1-9

Abstract:In this paper an interesting and famous realistic Lorenz's nonlinear problem is discussed using the Adomian Decomposition Method (ADM). The results (approximate solutions) obtained very accurate using classical Runge-Kutta (RK) method, single-term Haar Wavelet series [8] and ADM methods are compared with the ODE45 in Matlab. It is found that the solution obtained using ADM is closer to the ODE45 in Matlab. The high accuracy and the wide applicability of ADM approach will be demonstrated with numerical example. Solution graphs for discrete exact solutions are presented in a graphical form to show the efficiency of the ADM. The results obtained show that ADM is more useful for solving Lorenz's nonlinear problems and the solution can be obtained for any length of time.

[1] G. Adomian, "Solving Frontier Problems of Physics: Decomposition method", Kluwer, Boston, MA, 1994.
[2] J. C. Butcher, "The Numerical Methods for Ordinary Differential Equations", John Wiley & Sons, U.K., 2003.
[3] S. Sekar and A. Kavitha, "Numerical Investigation of the Time Invariant Optimal Control of Singular Systems Using Adomian Decomposition Method", Applied Mathematical Sciences, vol. 8, no. 121, pp. 6011-6018, 2014.
[4] S. Sekar and A. Kavitha, "Analysis of the linear time-invariant Electronic Circuit using Adomian Decomposition Method", Global Journal of Pure and Applied Mathematics, vol. 11, no. 1, pp. 10-13, 2015.
[5] S. Sekar and M. Nalini, "Numerical Analysis of Different Second Order Systems Using Adomian Decomposition Method", Applied Mathematical Sciences, vol. 8, no. 77, pp. 3825-3832, 2014.
[6] S. Sekar and M. Nalini, "Numerical Investigation of Higher Order Nonlinear Problem in the Calculus of Variations Using Adomian Decomposition Method", IOSR Journal of Mathematics, vol. 11, no. 1 Ver. II, (Jan-Feb. 2015), pp. 74-77.

Abstract: Let 𝑅 be a 2-torsion free semiprime ring. Let 𝐷 .,. :𝑅×𝑅→𝑅 be a symmetric left bi-derivation such that if (i) 𝑥𝑦±𝑑 𝑥𝑦 =𝑦𝑥±𝑑(𝑦𝑥), for all 𝑥,𝑦∈𝑅 and (ii) 𝑥,𝑦 −𝑑 𝑥𝑦 +𝑑(𝑦𝑥)∈𝑍(𝑅) or 𝑥,𝑦 +𝑑 𝑥𝑦 −𝑑(𝑦𝑥)∈𝑍(𝑅) for all 𝑥,𝑦 ∈𝑅, where 𝑑 is a trace of 𝐷. Thenboth the cases of 𝑅 is commutative. Key Words: Semiprime ring, Symmetric mapping, Trace,Derivation, Symmetric bi-derivation, Symmetric left bi-derivation.

[1]. Asharf. M.: On symmetric bi-derivations in rings, Rend. Istit. Mat. Univ. Trieste, Vol. XXXI, (1990), 25 – 36.
[2]. Daif. M. N and Bell. H. E.: Remarks on derivations on semiprime rings, Internat. J. Math.& Math. Sci. 15 (1992), 205 – 206.
[3]. Maksa.Gy.: A remark on symmetric bi-additive functions having nonnegative diagonalization, Glasnik Mat. 15 (35), (1980), 279 – 282.
[4]. Maksa. Gy.: On the trace of symmetric bi-derivations, C. R. Math. Rep. Acad. Sci.Canada 9 (1987), 303 – 307.
[5]. Posner. E. C.: Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093 – 1100.
[6]. Vukman. J.: Symmetric bi-derivations on prime and semi prime rings, Aequationesmathematics, 38 (1989), 245 – 254.
[7]. Vukman. J.: Two results concerning symmetric bi-derivations on prim rings, Aequationes Math.40 (1990),181 – 189.

Abstract:An antimagic labeling of a digraph D with p vertices and q arcs is a bijection f from the set of all arcs to the set of positive integers f:{1, 2, 3, …,q} such that all the p oriented vertex weights are distinct, where an oriented vertex weight is the sum of the labels of all arcs entering that vertex minus the sum of the labels of all arcs leaving it. A digraph D is called antimagic if it admits an antimagic סּlabeling. In this paper we investigate the existence of antimagic labelings of symmetric digraphs using Skolem sequences.

Keywords: antimagic labeling, symmetric digraph. 2010 Mathematics Subject Classification: 05C78.

[1]. Charles, J. Colbourn and Jeffrey , H. Dinitz, Handbook of combinatorial designs, Chapman & Hall/ CRC, 2nd edition, 2006.
[2]. G. Chartrand and L. Lesniak, Graphs and Digraphs, Chapman and Hall, CRC, 4th edition, 2005.
[3]. D. HefeTz, Torsten Mutze and Justus Schwartz, On Antimagic Directed Graphs. J. Graph Theory, 64 (2010), 219--232.
[4]. M. Nalliah, Antimagic total labelings of graphs and digraphs, Ph.D., Thesis, Kalasalingam University, India (2014).
[5]. N. Shalaby, Skolem sequences: Generalizations and Applications, Ph.D., Thesis, McMaster University, Canada(1992).

Abstract:In this paper, He's Homotopy Perturbation Method (HHPM) is used to study the linear first order fuzzy differential equations (FDE). The results obtained using He's Homotopy Perturbation Method and the methods taken from the literature [9] were compared with the exact solutions of the linear first order fuzzy differential equations. It is found that the solution obtained using the He's Homotopy Perturbation Method is closer to the exact solutions of the linear first order fuzzy differential equations. Error graphs for discrete and exact solutions are presented in a graphical form to highlight the efficiency of this method.
Keywords: Fuzzy Differential Equations, Differential Equations, Initial Value Problems, He's Homotopy Perturbation Method, Leapfrog Method.

[1] S. Abbasbandy and T. Allahviranloo, Numerical solutions of fuzzy differential equations by Taylor method, Journal of Computational Methods in Applied Mathematics. 2, 2002, 113-124.
[2] S. S. L. Chang and L. A. Zadeh, On fuzzy mapping and control, IEEE Transactions on Systems, Man, and Cybernetics, 2, 1972, 30–34.
[3] D. Dubois and H. Prade, Towards fuzzy differential calculus.III. Differentiation, Fuzzy Sets and Systems, 8( 3), 1982, 225–233.
[4] M. L. Puri and D. A. Ralescu, Differentials of fuzzy functions, Journal of Mathematical Analysis and Applications, 91(2), 1983, 552–558.
[5] R. Goetschel and Woxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18, 1986, 31-43.
[6] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24(3), 1987, 301–317.

Abstract: Until now, it was not known if this constant is irrational, let alone transcendental (Wells 1986, p. 28). The famous English mathematician G. H. Hardy is alleged to have offered to give up his Savilian Chair at Oxford to anyone who proved 𝛾 (the Euler-Mascheroni constant) to be irrational (Havil 2003, p. 52), although no written reference for this quote seems to be known. Hilbert mentioned the irrationality of 𝛾 as an unsolved problem that seems "unapproachable" and in front of which mathematicians stand helpless (Havil 2003, p. 97). This paper has been written to prove the trascendency of 𝛾. Keywords:Euler-Mascheroni constant, gamma, Harmonic series, trascendental numbers, Lindemann-Weierstrass theorem, Mertens' 3rd theorem

[1]. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 28, 1986.
[2]. Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.
[3]. "Euler-Mascheroni Constant" - http://mathworld.wolfram.com/Euler-MascheroniConstant.html
[4]. "Transcendental number" - https://en.wikipedia.org/wiki/Transcendental_number
[5]. "Irrational number" - https://en.wikipedia.org/wiki/Irrational_number
[6]. "Lindemann-Weierstrass theorem" - https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem

Abstract: Popoviciu ( 1935) proved his result for Bernstein Polynomials. We tested the degree of approximation of function by our newly defined Bernstein type Polynomials, and so the corresponding results of Popoviciu have been extended for Lebesgue integrable function in by our newly defined Bernstein type Polynomials

[1]. Anwar Habib (1981)."On the degree of approximation of functions by certain new Bernstein typePolynomials". Indian J. pure
Math.,12(7):882-888.
[2]. Anwar Habib & Saleh Al Shehri(2012) "On Generalized Polynomials I " International Journal of Engineering Research and Development e-
ISSN:2278-067X, 2278-800X, Volume 5, Issue 4 ,December 2012 , pp.18-26
[3]. Anwar Habib; (2015)" On Bernstein Polynomials" IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X.
Volume 11, Issue 1 Ver. V (Jan - Feb. 2015), PP 26-34 www.iosrjournals.org
[4]. Cheney, E.W. , and Sharma, A.(1964)."On a generalization of Bernstein polynomials".Rev. Mat. Univ. Parma(2),5,77-84.
[5]. Jensen, J. L. W. A. (1902). "Sur uneidentité Abel et surd‟autressformulesamalogues".Acta Math. , 26, 307-18

Abstract: The generalized fractional Hilbert transform plays an important role in signal processing, image reconstruction, etc. This paper generalizes the fractional Hilbert Transform to the spaces of generalized functions and proved analyticity theorem also obtained many operation formulae for the transform. Key words: Hilbert transform, generalized fractional Hilbert Transform, Signal Processing.

[1] Bhosale, B. N., Chaudhary, M. S., Bull. Cal. Math Soc., 94(5), 349, 2002.
[2] Namias, V. J. Inst. Math Appl, 25, 241, 1980.
[3] Tatiana Alieva and Bastiaans Martin J.: "On Fractional Fourier transform moments", IEEE Signal processing Letters, Vol. 7, No. 11,
Nov. 2000.
[4] Tatiana Alieva and Bastiaans Martin J.: "Wigner distribution and fractional Fourier transform for 2- dimensional symmetric beams",
JOSA A, Vol. 17, No. 12, Dec. 2000, pp. 2319-2323.

Abstract: The unsteady free convective viscous incompressible flow of an electrically conducting fluid, under uniform magnetic field, through porous medium near vertical insulated wall with variable suction in slip flow has been investigated. The governing equation in non-dimensional form is solved with the help of Perturbation technique. An approximated solution for velocity, skin friction, temperature and Nusselt number are obtained. The effects of various non-dimensional parameters on velocity profile, skin friction and Nusselt number are shown graphically.

[1]. Berezovsky, A. A., Martynenko, O. G. and Yu, A., Sokovishin, Free convective heat transfer on a vertical semi infinite plate, J.
Engng. Phys., 33, 32-39, 1977.
[2]. Chandran, P., Sacheti, N. C., and Singh, A. K., Effects of rotation on unsteady hydrodynamic Couette flow, Astrophysics and Space
Science, 202, 1–10, 1993.
[3]. Chandran, P., Sacheti, N. C., and Singh, A. K. Haydromagnetic flow and heat transfer past a continuously moving porous boundary,
International Communication in Heat and Mass Transfer, 23, 889–898, 1996.
[4]. Chandran, P., Sacheti, N. C., and Singh, A. K. Unsteady hydromagnetic free convection flow with heat flux and accelerated
boundary motion, Journal of Physical Society of Japan, 67, 124–129, 1998.
[5]. Chandran, P., Sacheti, N. C., and Singh, A. K. An undefined approach to analytical solution of a hydromagnetic free convection
flow, Scientiae Mathematicae Japonicae, 53, 467–476, 2001.