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.

[1]. Besag, J., 1977 "Errror-in-variables estimation for Gaussian schemes. Journal Royal Statistical Society, B 39, 73-78.
[2]. Gill, P.S., and Shukla, G.K., 1985 "Efficiency of Nearest Neighbour Balanced Block Designs for Correlated Observations," Biometrika. 72, 639 -644.
[3]. Kunert, J., 1987, "Neighbour Balanced Block Designs for Correlated Errors. Biometrika. 74, 4, 717 – 724.
[4]. Jin, B. and Morgan, J.P. (2008) " Optimal Saturated Block Designs when observations are correlated, J. Statist. Plann.Infernce, in press, doi:10.1016/j.2006.06.048.
[5]. Lee, K.Y., And Jacroux, M., 1987,"Some Sufficient Conditions Fore-Optimality and Mv-Optimality of Block Designs having Blocks of Unequal Size,"Ann.Inst.Statist.Math. 39, 385 – 397.
[6]. Martin, R. J. and Eccleston, J.A. (1991), " Optimal incomplete Block Designs for General Dependence Structures. J. Statist. Plann.Inference., 28, 68 -81.
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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.
[10]. Santharam, C., Ponnusamy, K.N and Chandrasekar, B., 1996,"Universal Optimality Of Nearest Neighbour Balanced Block Designs using ARMA Model,". Biometrical.J. 38, 725 – 730.

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.

[1]. N.K. Bhadra, IOSRJM, H0244145, The Complex Model of the Universe, Issue 4, 41 (2012).
[2]. N.K. Bhadra, IOSRJM, D0412033, The Complex Model of the Quantum Universe, vol.4, Issue 1, 20 (2012).
[3]. A.Y. Kamenshchik and Serena-Manti, Phys Rev D85, 123518 (2012).
[4]. C.W. Misner, Phys Rev Lett 22, 1071 (1969).
[5]. B.S. DeWitt, Phys Rev 160, 1113 (1967).
[6]. C. Kiefer, Quantum Gravity (Oxford University Press, Oxford, 2007), 2nd ed.
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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.

Abstract: Given a set of polynomials 𝐹⊆𝑘 𝑋 , what is the dimension of the affine variety 𝐕 𝐹 ? Giving an affirmative answer to this question was never so easier in algebraic geometry until the development of Groebner basis. Groebner bases are nice because we can compute these; and this is Buchberger algorithm which makes Groebner bases so fruitful. Given a polynomial collection 𝐹⊆𝑘 𝑋 , Groebner basis helps in computing free set in 𝑘 𝑋 with respect to the ideal 𝐹 . Then dimension of 𝐕 𝐹 equals the cardinality of the free set with respect to 𝐹 . Here we describe an algorithm for computing such free set and answer the aforementioned question from computational point of view. We see that, in case of finding dimension of affine variety, computational technique is much more informative and motivating than theoretical method.

Keywords: Affine variety, Dimension, Free set, Groebner basis

[1] D. Cox, J. Little and D. O'Shea, An introduction to computational algebraic geometry and commutative algebra (New York Inc, Springer-Verlage, 1997).
[2] Mohammad Salah Uddin, Computational algebraic geometry and Groebner bases, Diploma dissertation, The Abdus Salam International Center for Theoretical Physics, Trieste, Italy, 2009.
[3] B. Sturmfels, Lectures on Groebner bases, Mathematical Science and Research Institute, Berkeley.
[4] Karen E. Smith, Lauri Kahanpaa, Pekka Kekalainen, William Traves, An Invitation to algebraic geometry, (New York Inc, Springer-Verlage, 2000)

Abstract: The famous Fermat's Last Theorem was proved, after three and a half centuries, by Prof: Andrew Wiles and his associate Prof: Richard Taylor in 1994. It is highly advanced. There is search for a simple proof. Congruence modulo addition and multiplication theorems, which are common text book matters, comes to our help. This proof, if found valid, offers very simple one that can be understood by UG students as well.

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Abstract: The basic definitions of fuzzy independent set, fuzzy dominating set and fuzzy independent dominating sets are discussed. In this paper we introduce the concept of perfect domination in fuzzy graphs and obtain some interesting results for this new parameter in fuzzy graphs and the aim of this paper is to find on what relations the fuzzy graph has perfect domination number and independent domination number. Finally, the independent domination number for a connected fuzzy graph is obtained.

Key words: Fuzzy graph,Fuzzy dominating set, fuzzy independent domi- nating set, perfect dominating set

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[4]. Carrington, J.R., Harary, F., and Haynes, T.W., Changing and un- changing the domination number of a graph. J.Combin., Math. Com- bin. Comput., 9: 57-63, 1991.

Abstract: The main purpose of this paper is to obtain fixed point theorems for sequence of mappings in strict contractive conditions which generalizes Theorem 1 of Aamri [1].

Key words and phrases:: Fixed point, Coincidence point, compatible maps ,weakly compatible map, non-compatible maps, property(E.A).

[1] M. Aamri and D. El. Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Theory and App., 270 (2002) 181-188.

[2] U.karuppiah and M.Marudai, common fixed point theorems for sequence of mappings under strict contractive conditions,

[3] G. Jungck, Compatible mappings and common fixed points, Intl. J. Math. Sci., 9 (1986) 771-779.

[4] J. Jachymski, Common fixed point theorems for some families of maps, Ind. J. Pure Appl.Math., 25 (1994) 925-937.

[5] G. Jungck, K.B. Moon, S. Park and B.E. Rhodes, On generalisation of the Meir-Keeler type contraction maps corrections, J. Math. Anal. Appl., 180 (1993) 221-222.

[6] R.P. Pant, Common fixed points of sequences of mappings, Ganita, 47 (1996) 43-49.

[7] R.P. Pant, Common fixed points of contractive maps, J. Math. Anal. Appl., 226 (1998) 251-258.

[8] S. Sessa, On a weak commutativity condition of mappings in fixed point considerations, Publ.Int. Math. (Belgrad), 32 (1982), 149-153.

Abstract:This study applies Dickey-Fuller test for unit root to import and export data on seaborne trade figure in Nigeria. The Dicky-Fuller test shows that both data series are non-stationary. First differencing of the data stationarized it and ordinary least squares regression analysis applied to the data shows a significant relationship between export and import trade in Nigeria.. the graph of the data shows that there was an upward trend in both import and export. Evaluation statistics of tau, R- square and F- ratio were used for the estimated model.

Key words: Import, Export, Seaport, unit root test and Dickey - Fuller.

[1]. Achike, A. I, Onoja, A. O. and Herbert, B. C. (2011). Analysis of Determinants of Agricultural Foreign Direct Investment in Nigeria: The Nigerian Agricultural Journal vol.42 pp 1- 11
[2]. Bannerjee, Anindya, Junan Dolado, John W. Galbraith and David F. Hendry (1993) Integration, Error Correction and the Econometric Analysis of Non- Stationary Data Oxford: Oxford University Press.
[3]. Box, G.E.P. and G.M. Jenkins (1979) Time series Analysis Forecasting and Control. San Francisco: Holden Day.
[4]. Central Bank of Nigeria (2006). "Annual Report on Nigeria Port Plc" pp 47- 50.
[5]. Danilowska, Alina (2008). Preferential Macroeconmic Determinants of Agricultural Preferential Investmen Credit in Poland. A Paper Presented at 2008 International Congress. August 26-29 Ghent Belgium. Retrieved 25th November, 2011 from http://purl. umn.edu/4406.
[6]. Dickey, David A., and Wayne A. Fuller. 1979. Distribution of the Estimators for Autoregressive Time Series with a Unit Root. Journal of the American Statistical Association 74: 427-431.
[7]. Engle, R.F. and C.W.J Granger , (1987). Cointegration and Error Correction: Representation,Estimation and Testing. Econometrica vol. 55: pp 251-276
[8]. Granger, C. W. J. and Newbold, P. (1974)Spurious Regression in Econometrics. Journal of Econometrics, vol.2 (2) pp 111- 120.
[9]. Greene,W. H. (2008) Econometric Analysis Fifth edition Prentice Hall: New Jersey.
[10]. Gujarati, D. N. (2006), Essentials of Econometrics, 3rd ed. McGraw-Hill, New York.

Abstract: The main aim of this article is to present the Spectral theory of self-adjoint operators on Hilbert space and to describe its applications in the development of Quantum mechanics. Since in Quantum mechanics, observables correspond to self-adjoint operators, to achieve our aim, we employed the idea of symmetric operators in Hilbert space. We see that there is a bijection between symmetric extensions of an operator and isometric extensions of its Cayley transform. The resolvent (a) is invertible, which is the generalization of the theory of eigenvalues of a matrix, that is, the set of all eigenvalues of an operator is called the spectrum of the operator, and more importantly, the Hermitian operator. Finally, we showed how to determine the energy level of the atom of an element using some derived equations.

Keywords: Symmetric operator, Self-adjoint operator, Observables, Eigenvalues, Energy

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Abstract: In a linear motion of a system of two satellites connected by extensible cable, one stable equilibrium point exists when perturbative forces like Solar radiation pressure, shadow of the earth, oblateness of the earth, air resistance and earth's magnetic force act simultaneously. We have obtained one stable point of equilibrium in case of perturbative forces like the shadow of the earth due to solar radiation pressure, Magnetic Force and oblateness of the earth acting together on the motion of a system of two satellites connected by extensible cable in the central gravitational field of earth in case of circular orbit of the centre of mass. We have used Liapunov's theorem on stability to examine the stability of the equilibrium point.

Key words: Stabiity, Equiliribium Point, Solar Radiation Pressure, circular orbit, Liapunov Theorem, Satellites

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[5] V. Kumar and N. Kumari, Stability of Equilibrium point of the centre of mass of an extensible cable connected
satellites system in case of circular orbit in three dimensional, IJSER,Vol-4,Issue 9, (2013),1802-1808

Abstract: The smoothness-increasing operator "convolution" is well known for inheriting the best properties of each parent function. It is also well known that if f  L1 and g is a Bounded Variation (BV) function, then f  g inherits the properties from the parent's spaces. This aspect of BV can be generalized in many ways and many generalizations are obtained. However, in this paper we introduce the notion of p – Bounded variation function. In relation to that we show that the convolution of two functions f  g is the inverse Fourier transforms of the two functions. Moreover, we prove that if f, g  BV(p)[0, 2], then f  g  BV(p)[0, 2], and that on any locally compact Abelian group, a version of the convolution theorem holds.

Keywords: Convolution, p-Bounded variation, Fourier transform, Abelian group.

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Abstract: In this paper, we present a continuous block integrator for direct integration of stiff and oscillatory first-order ordinary differential equations using interpolation and collocation techniques. The approximate solution used in the derivation is a combination of power series and exponential function. The paper further investigates the properties of the block integrator and found it to be zero-stable, consistent and convergent. The integrator was also tested on some sampled stiff and oscillatory problems and found to perform better than some existing ones.

Keywords: Approximate Solution, Block Integrator, Continuous, Oscillatory, Stiff AMS Subject Classification (2010): 65L05, 65L06, 65D30

[1] I. J. Ajie, M. N. O. Ikhile, and P. Onumanyi, "A Family of A( )  Stable Block Methods for Stiff ODEs", Journal of
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Abstract: In this paper, we use new approach for linear factorized quadratic optimization and a quadratic fractional optimization problem. We observed that there is change in the rule of selecting entering vector at initial stage and for some quadratic optimization problem; it takes more number of iteration to achieve optimality. Here at the initial step we choose the entering vector on the basis of new rules of method described below. Like a linear fractional programming problem (LFPP), linear factorized quadratic optimization problem (LFQOP) and quadratic fractional optimization problem (LFQFOP) can be usefully applied in a wide range of real-world applications. These are useful in solving the problem in economics whenever the different economic activities utilize the fixed resources in proportion to the level of their values, hospital and health care planning, financial planning etc. In the last few decades a lot of research papers and monographs were published throughout the world where authors investigated different theoretical and algorithmic aspects of QOP and QFO problems in various forms. Quadratic fractional program is an optimization problem wherein one either minimizes or maximizes a quadratic fractional objective function subject to finite number of linear inequality or equality constraints. In this paper, we propose solution methods for linear factorized quadratic optimization problem and factorized quadratic fractional optimization problem with new approach.

Keywords: Optimality Conditions, LFQOP, LFQFOP, Simplex Technique.

[1]. Abdulrahim B. K. (2013), Solving Quadratic Fractional Programming Problem via Feasible Direction Development and Modified Simplex Method. Journal of Zankoy sulaimani- part A (JZS-A), 15(2), pp 45-52.
[2]. Fukushima M. and etc. all (2008), Quadratic Fractional Programming Problems with Quadratic Constraints, Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University.
[3]. Hasan M.B. (2012), A technique for solving special type quadratic programming problems, Dhaka university j. sci. 60(2) pp-209-215.
[4]. Khobragade N. W. and ets all (2012), alternative approach to wolfe's modified simplex method for quadratic programming, Int. J. Latest Trend Math vol.-2(1) pp-19-24.
[5]. R. Enkhbat and etc. all (2011), A method for fractional programming, International Journal of Pure and Applied Mathematics, Volume 73 No. 1, pp.93-99.
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Abstract: In this paper, we aimed at determining all subgroups of the Symmetric group S5 up to Automorphism class using Sylow's theorem and Lagrange's theorem. This is achieved by finding all subgroups of order m for which m|O(S5) and are subsets of S5. It was vividly described and derived 156 subgroups of S5 and their conjugacy class size and Isomorphism class. The Alternating group A5 is the unique maximal normal subgroup of S5. Further, the Symmetric group S5 is centerless and every automorphism of it is inner. Also, every natural homomorphism to the automorphism group is an isomorphism. Hence, S5 is complete. The derived subgroups can be used to determine the number of Fuzzy subgroups of the symmetric group S5 for further research.

Keywords: Symmetric group, Conjugacy class, Isomorphism, Automorphism, Complete group

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[5] D. Samaila , M. Pius Pur and B. Ibrahim Abba, Visualizing the Homorphic Image Through Abstract Characterization of the Symmetry Group Dn, Int. J. Pure Appl. Sci. Technol., 15(2), 2013, pp. 33-42.

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