iosr-Jm   Volume-8 ~ Issue-3

Abstract: Here a particular method is made to generate a A Single Formula to find the nth term and sum of n terms of first n Kth dimensional S sided Polygonal numbers.

Keywords: Dimensional Polygonal Numbers, Polygonal Numbers,Square Numbers, Triangular Numbers, 3Dimensional Polygonal Numbers,

[1]. MATHEMATICS for the MILLION - Lancelot Hogben 1978

Abstract: Block designs for observations correlated in one dimension are investigated. Santharam and Ponnusamy (1995, 1996) investigated the universal optimality on Nearest Neighbor Balanced Block Designs (NNBD) using first order and second order correlated models (AR(1), MA(1) , ARMA (1,1) and AR (2), MA(2)). Ruban raja and santharam (2013) investigated the MV-optimality of Nearest Neighbour Balanced Block Designs using AR(1), MA(1) and ARMA (1,1) ( First order Auto Regressive, First order Moving Average and First order Auto Regressive Moving average) model for five treatments. In this paper we have investigated MV-optimality of Nearest Neighbour Balanced Block Designs using AR(2) and MA(2) ( Second order Auto Regressive and Second order Moving Average) models for five treatments

Key words: Auto-regressive Model, Moving Average model, MV-optimality, Optimal experimental design.

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[8]. Ruban Raja B., Santharam C., and Ramesh kumar, 2012, "MV – Optimality of Nearest Neighbour Balanced Block Designs using First order and Second order Correlated models," International Journal of Statistika and Mathematika( ISSN: 2277 – 2790 E-ISSN: 2249 – 8605)
[9]. Ruban Raja B., Santharam C., 2013, "MV – Optimality of Nearest Neighbour Balanced Block Designs using First order Correlated models," International Journal of Statistics Analysis, ISSN 2248-9959 Vol.3, pp . 379 -389.
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Abstract: We study a relation between the cosmological singularities in classical and quantum theory, comparing the classical and quantum dynamics in three models possessing the Big-Rip Singularity : the model mainly based on tachyon field and pseudo-tachyon field with respect to the pseudo-space R iR I ( , where 'R' is the scale factor of the universe and i  1 ). It was shown that in the pseudo-tachyon model the Hamiltonian is well defined and hence wave function of the universe is not obliged to vanish at the values of the variables, corresponding to the scale factor RI of the appearance of the Big-Rip singularity. There is some kind of a classical-quantum correspondences in the classical cosmological tachyon model exists an infinite oneparameter set of the cosmological evolutions encountering and crossing the Big-Rip singularity. In all other cases—mainly the Big-Brake and Big-Bang singularity in the scalar field model and the Big-Crunch and the Big-Rip singularities in both the tachyon and pseudo tachyon field model we have observed the phenomenon of the quantum avoidance of singularities. It corresponds to the degeneration of the corresponding cosmological trajectories in classical theory. It was shown that a negative pressure may be acquired from Big-Rip Singularity, which is responsible for the Big-Bang singularity and then the expansion of the universe.

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Abstract: Intrigues most researchers about the Riemann zeta hypothesis is the ability to employ cum different approaches with instinctive mindset to obtain some very interesting results. Motivated by their style of reasoning, the result obtained in this work of redefining or re-representation of Riemann zeta function in different forms by employing different techniques on two functional equations made the results better, simpler and concise new representations of Riemann zeta function.

Keywords: Analytic Continuation, Osborne's rule, Riemann Zeta Function

[1] E. Bombieri, (2000), Problems of the millennium: The Riemann hypothesis, Clay mathematical Institute.
[2] O. Enoch, (2012), A New Representation of the Riemann Zeta Function Via Its Functional Equation
[3] O. Enoch, (2012), A General Representation of the Zeros of the Riemann Zeta Function via Fourier series Expansion
[4] O.O.A. Enoch and F.J. Adeyeye, (2012), A Validation of the Real Zeros of the Riemann Zeta Function via the Continuation Formula of the Zeta Function, Journal of Basic & Applied Sciences, 2012, 8, 1
[5] P. Sarnak, (2004), Problems of the Millennium: The Riemann Hypothesis by Princeton University and courant Institute of Mathematics.