iosr-Jm   Volume-10 ~ Issue-3 ~ May-June 2014

Abstract: This paper discusses the application of matrix-vector operations technique for solving constrained optimization problems. The method is aimed at circumventing the computational rigours undergone using the simplex and revised simplex method in solving this class of problems with aim of reducing the computer memory space occupied by the methods. In other to achieve this, a straightforward and simple to handle matrix-vector operations algorithm has been developed to solve the same constrained problems. Numerical results show some improvements compared with the classical method.
Keywords: Constrained Optimization Problem, Vector Operations and Matrix Operations

[1]. RAO, S. S., (1978), Optimization Theory and Applications, Willy and Sons.
[2]. THOMAS, F.E., and DAVID, M.H.,(2001), Optimization of Chemical Processes, McGraw Hill
[3]. OKONTA, P. N. and J. S. APANAPUDOR,(2001), Undergraduate Linear Algebra, Functional Publishing Company, Surulere, Lagos State. Nigeria.
[4]. RICHARD BRONSON and GOVINDASAMA NAADIMUTHU.(2007), Operations Research, Tata McGraw-Hill Publishing Company Limited, New Delhi.
[5]. SEYMOUR LIPSCHUTZ and MARC LIPSON,(2009), Linear Algebra, McGraw-Hill Companies, New Delhi.
[6]. GUPTA P. K. and HIRA D. S. (2012), Operations Research, S. Chand & Company Ltd, New Delhi

Abstract: This paper discusses the application of a Modified Lagrange Multipliers Method (MLM) in solving optimization problems with equality and inequality constraints. The method is aimed at circumventing the computational rigours undergone using the Lagrange multipliers method in solving this class of problems with equality and inequality constraints independently. Also, it aims at reducing the computer memory space occupied by the independent methods in solving these problems using the said methods.
In other to achieve this, a straightforward and simple to handle MLM algorithm has been developed to solve the same optimization problems with both equality and inequality constraints. Comparing the numerical results with that of the classical methods show some improvements.
Keywords: Lagrange Multipliers, Modified Lagrange Multipliers, Optimization Problems with Equality Constraints, Optimization Problems with Inequality Constraints

[1]. RAO, S. S., (1978), Optimization Theory and Applications, Willy and Sons.
[2]. THOMAS, F.E., and DAVID, M.H., (2001), Optimization of Chemical Processes, McGraw Hill
[3]. IGOR GRIVA STEPHEN G and NASH ARIELA SOFER (2009), Linear and Nonlinear Optimization, George Mason University Fairfax, Virginia.
[4]. JAN A. SNYMAN (2005), Practical Mathematical Optimization, University of Pretoria, Pretoria, South Africa.
[5]. RICHARD BRONSON and GOVINDASAMA NAADIMUTHU.(2007), Operations Research, Tata McGraw-Hill Publishing Company Limited, New Delhi.

Abstract: In this paper, we constructed a control operator, G, which enables an Extended Conjugate Gradient Method (ECGM) to be employed in solving for the optimal control and trajectories of continuous time linear regulator problems. Similar operators constructed in the past by various authors have limited application. This call for the construction of the control operator that is aimed at taking care of any of the Mayer's, Lagrange's and Bolza's cost form of linear regulator problems. The authors of this paper desire that, with the construction of the operator, one will circumvent the difficulties undergone using the classical methods and its application will further improve the result of the Extended Conjugate Gradient Method in solving this class of optimal control problem.
Keywords: Continuous Linear Regulator Problem, Control Operator, Extended Conjugate Gradient Method and Optimal Control.

[1]. Athans, M. and Falb, P. L., (1966), Optimal Control: An Introduction to the Theory and Its Applications, McGraw-Hill, New York.
[2]. Aderibigbe, F. M., (1993), "An Extended Conjugate Gradient Method Algorithm for Control Systems with Delay-I", Advances in Modeling & Analysis, C, AMSE Press, Vol.36, No. 3, pp51-64.
[3]. George M. Siouris, (1996), An Engineering Approach To Optimal Control And EstimationTheory,John Wiley & Sons, Inc., 605 Third Avenue, New York, 10158-00 12.
[4]. Hasdorff, L. (1976), Gradient optimization and Nonlinear Control. J. Wiley and Sons,NewYork.
[5]. Ibiejugba, M. R. and Onumanyi, P., (1984), "A Control Operator and some of its Applications, J. Math. Anal. Appl. Vol. 103, No. 1. Pp. 31-47.

Abstract: The acceptability of an algorithm is a function of its implementability and convergence. In this paper, we examine some features of the extended conjugate gradient method (ECGM) algorithm, one of the optimization techniques for solving continuous/or discrete optimal control problems It is observed while using this algorithm, there is a consistent demand for some of the features of the algorithm. Among these are the stepsize, alpha, the gradient(the partial derivatives), the search directions e.t.c. One of these features closely examined is the computation of J , the gradient of J , the performance index , , ( , )T J  z Hz z  x u , which is foremost while implementing the algorithm. In the light of this, we develop an explicit expression for J .Furthermore, a generalization of the expression for J , for all positive integers n was attained, via mathematical induction.
Keywords: Step-size, Operator,Conjugate Search Directions

[1]. Griffel, D.H.(1973). Applied functional Analysis, Kills Horwood Limited Chichester.
[2]. Ibiejugba, M.A.(1980) Computing Methods in Optimal Control, Ph. D Thesis University of Leeds, England.
[3]. Ibiejugba, M.A. (1985) Computing Gradient descent of a Smooth functional, Advances in Modeling and Simulation, Vol.4.pp. 37-
43.
[4]. Macki, J. and Straus, A.(1980) Introuction to Optimal Control Theory, Springer-Verlag, N.Y. Inc
[5]. Oliviera, I.B.(2002) A "HUM" Conjugate Gradient Algorithm for Constrained Non-linear
[6]. Optimal Control: Terminal and Regulator Problems, APh.D Thesis presented to the Institute of Technology, Massahusetts.

Abstract: In this paper, we examine techniques for the construction of the conjugate search direction, xi p and ui p ,often required in the implementation of the Extended Conjugate Gradient Method(ECGM) for optimal control problems. The various techniques were derived analytically usingsome ideas from numerical linear algebra. We also establish the authenticity of these approaches by presenting a proof via mathematical induction, which when applied for the computation of these vectors proved successful most especially for the Discrete Optimal Control Problems(DOCP).
Keywords: Conjugate Search Direction, Operator, Parameter

[1]. Aderibigbe, F.M. (1988):An Extended Conjugate Gradient Method Algorithm for EvolutionEquations, Ph.D Thesis,
University of Ilorin,Nigeria.
[2]. Aderibigbe, F.M.(1995):An Extended Conjugate Gradient Method Algorithm for Control Systems with Delay - I, Advances
Modeling and Analysis,C, AMSE Press Vol. 36,No.3,pp 51-65.
[3]. Grffel, D.H.(1993) Applied functional Analysis, Horwood Limited Chichester.
[4]. Hager, W. W.and Zhang, H.(2005) A survey of nonlinear conjugate gradient methods, Journal ofAmerican Mathematical Society,
Chicago.
[5]. Hager, W. W. and Zhang, H(2003) A new conjugate gradient method with guaranteed descentand an efficient line search, SIAM J.
Optimization

Abstract: In this paper, approximation techniques based on the shifted Jacobi together with spectral tau technique are presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain. The fractional derivatives are described in the Caputo sense. The technique is derived by expanding the required approximate solution as the elements of shifted Jacobi polynomials. Using the operational matrix of the fractional derivative, the problem can be reduced to a set of linear algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the technique and a comparison is made with the existing results to show that the proposed method is easy to implement and produce accurate results. Keywords: Fractional calculus, Jacobi polynomials, Tau method, Operational matrix.

[1] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
[2] D. Baleanu, J.H. Asad, I. Petras, fractional Euler-Lagrange Equation of Caldirola-kanai Oscilator, Rommanian Reports on Physics 64,
90 7 (2012).
[3] M. Dalir, M. Bashour, Applications of Fractional Calculus, Applied Mathematical Sciences 4, 1021(2010).
[4] S.Das, Functional Fractional Calculus for System Identification and Controls (Springer, New York, 2008).
[5] I. Podlubny, Fractional Differential Equations (Academic Press Inc., San Diego, CA, 1999).
[6] H.Jafari, M. Saeidy, D. Baleanu, The variational iteration method for solving n-th order fuzzy differential equations, Cent.
Eur.J.Phys.10,76 (2012).

Abstract: The conjugate gradient method is an algorithm for the numerical solution of systems of linear equations whose matrices are positive-definite and symmetric. It is an iterative method that can even be applied to solve a sparse system of equations. In this paper, we applied the conjugate gradient method of higher - order to wave propagation control problems. The algorithm of this method was implemented using MATLAB 7.10.0 codes to get numerical results.

Keywords: Wave Propagation, Optimal Control, Positive-definite matrix, Symmetric Matrix, Conjugate Gradient Method

[1]. Zakharenko, A.A. Thirty Two New SH-Waves Propagating in PEM Plates of Class 6 mm, Saarbruecken – Krasnoyarsk, LAP LAMBERT Academic Publishing GmbH & Co., 2012.
[2]. Banimok, J.A. and David K.R. An Iterative Method for Solving Wave Equations, PENTUM Journal of Mathematics and Computer Science, Vol. 5, 2006, 38-52.
[3]. Brillovin, L. Wave Propagation and Group Velocity, Academic press, New York, 1960.
[4]. Zakharenko, A.A. Seven New SH-Saws in Cubic Piezoelectromagnetics, Saarbruecken – Krasnoyarsk, LAP LAMBERT Academic Publishing GmbH & Co., 2011.
[5]. Kesavan, S. Topics in Functional Analysis and Applications, John willey, New York, 1989.

Abstract:The spread of communicable diseases is often described mathematically by compartmental models. A vaccine is a biological preparation that improves immunity to a particular disease. In this paper a nonlinear mathematical deterministic compartmental model for the dynamics of an infectious disease including the role of a preventive vaccine, natural birth rate and natural death rate is proposed and analyzed. The model has various kinds of parameter. We try to present a model for the transmission dynamics of an infectious disease and mathematically analyzed the stability of daisies free equilibrium and endemic equilibrium. Also we have given some strategy to control the epidemic by controlling the parameters.
Keywords: Stability analysis, Basic reproduction number, diseases free equilibrium, endemic equilibrium.

[1]. Kermack, W.O., & McKendrick, A.G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal
Society of London. Series A, 115, 700–721.
[2]. Driessche P.V.D. and Watmough J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models
of disease transmission. Mtahematical Bioscience, 180, 29-48.
[3]. Hethcote, H.W. (2000). The Mathematics of Infectious Diseases. Society for Industrial and Applied Mathematics, 42(4), 599-653.
[4]. Arino, J., McCluskey, C.C., & Driessche, P.V.D. (2003). Global results for an epidemic model with vaccination that exhibits
backward bifurcation. SIAM J. Appl. Math., 64, 260–276.
[5]. Buonomo, d'Onofrio, B. A., & Lacitignola, D. (2008). Global stability of an SIR epidemic model with information dependent
vaccination, Mathematical Biosciences, 216, 9–16.

Abstract: The object of the present investigation is to study the degree of approximation of Fourier series o function of bounded variation by generalized Harmonic-Cesaro method of summation.
Keywords: Fourier series, bounded variation

[1]. Bojanic, R.: An estimate of the rate of convergence for the Fourier series of function of bounded variation, publ. Inst Math
(Beograd) (N.S) 26(40), 1979 57-60
[2]. Bojanic, R and Mazhar, S.M.: An estimate of the rate of convergence of the Norlund voronoi means of the Fourier series of function
of bounded variation, Approx . Theory III,( Academic press 1980).
[3]. Bojanic, R. and Mazhar, S.M.: "An estimate of the rate of Convergence of Cesaro means of Fourier series of function of bounded
variation", Proc. of International Conf. Of Mathematics Analysis and its Application Kuwait, 1985, 17-22.
[4]. Das, G. and Mohapatra, P.C.: "On a generalized Harmonic-Cesaro method of summation and its application to Fourier series",
Indian Journal of
[5]. Sing, T.: "Degree of Approximation to functions in a normed space", Publ. Math. Debrecen, 40, 13-4, 1992, 261-267.
[6]. Zygmund, A.: Trigonometric series, second edition, Volume I and Volume II combined, (Cambridge University Press, 1933).

Abstract: In this paper we present a survey of three iterative refinement methods for the solution of system of linear equations. The result of this paper shows that the refinement of generalized Jacobi method is much more efficient than the refinement of Jacobi iterative method and is as fast as the refinement of Gauss-Seidel method considering their performance, number of iterations required to converge, storage and level of accuracy. This research will help to appreciate the use of iterative techniques for understanding linear equations. Keywords: Jacobi method, Gauss-Seidel method

[1]. Noreen Jamil, A Comparison of Direct and Indirect Solvers for Linear Systems of Equations. Int. J. Emerg. Sci., 2(2), 310-321, June-2012.
[2]. Ibrahim B. Kalambi, A Comparison of Three Iterative Methods for the Solution of Linear Equations, J. Appl. Sci. Environ. Vol. 12, 2008, no.4, 53-55.
[3]. Davod K. Salkuyeh, Generalized Jacobi and Gauss-Seidel Methods for Solving Linear Systems of Equations, Numer. Math. J. Chinese Uni (Englisher) issue 2, Vol.16: 164-170, 2007.
[4]. V.B. Kumer Vatti and Genanew Gofe Gonfa, Refinement of Generalized Jacobi (RGJ) Method for Solving System of Linear Equations, Int. J. Contemp. Math. Sciences, Vol.6, 2011, no.3, 109-116.
[5]. B. N. Dutta, Numerical Linear Algebra and Applications, Pacific Grove: California (1999).
[6]. D. C. Lay, Linear Algebra and its Applications, New York (1994).

Abstract: Herein is characterized the solution of quasiconvex optimization in nonlinear programming problem using Kuhn-tucker theorem for problems with inequality constraints. The purpose of this paper is to present sufficient conditions using Kuhn-Tucker first order conditions to identify optima solution of inequality constrained optimization problems.
Keywords: Qausiconvex, Nonlinear Programming, Optimization, Optimal Solution, Kuhn-Tucker, Convexity

[1]. Borwein, J.M. and Lewis, A.S. (2000): Convex Analysis and Nonlinear Optimization (Theory and Examples) Springer-Verlag, New York.
[2]. Boyd, S. and Vandersherghe, L. (2004): Convex Optimization, Cambridge University Press, United Kingdom.
[3]. Horst, R. et al (1990): Global Optimization (deterministic approaches), Springer-Verlag, New York.
[4]. Luenberger, D.G. (1969): Optimization by vector space methods. Wiley, New York.
[5]. Osinuga, I.A. and Bamigbola, O.M. (2004): On Global Optimization for Convex Problems on Abstract Spaces, ABACUS(J. Math. Ass. Nig) 31(2A), 94 – 98.
[6]. Osinuga, I.A. (2007): Further Results on the Classes of Convex Functions, J. Nig. Ass. Math. Phys. 11, 515 – 518.
[7]. Zalinescu, C. (2002): Convex Analysis in General Vector Spaces, World Scientific, Singapore.

Abstract: By a graph G = (V,E) we mean a finite, undirected graph without loops or multiple edges.The order and size of G are denoted by p and q respectively. For graph theoretical terms we refer to Harary [6] and for terms related to domination we refer Haynes et al.[7] A subset D of V is said to be a dominating set in G if every vertex in V  D is adjacent to at least one vertex in D. Kulli and Janakiram introduced the concept of nonsplit domination in graphs [9]. A dominating set D of a graph G is a nonsplit dominating set if <V  D > is connected. The nonsplit domination number (G) ns  of G is the minimum cardinality of a nonsplit dominating set. A nonsplit dominating set with cardinality (G) ns  is called a ns  -set. Acharya[1] introduced the concept of domsaturation number of a graph. The least positive integer k such that every vertex of G lies in a dominating set of cardinality k is called the domsaturation number of G and is denoted by ds(G) . A detailed study of this parameter was already done by Arumugam and Kala[2]. In this paper, we define nonsplit domsaturation number of a graph . We determine the value of this parameter for several< classes of graphs , obtain bounds for this parameter and also characterize the graphs which attain these bounds.

[1]. B. D. Acharya, The strong domination number of a graph and related concepts, J.Math. Phys.Sci. 14(1980), No. 5, 471-475.
[2]. S. Arumugam and R. Kala, Domsaturation number of a graph, Indian J. Pure appl. Math., 33(2002), No. 11, 1671-1676.
[3]. J. R. Carrington, F. Harary and T. W. Haynes, Changing and unchanging the domination number of a graph, J. Combin. Math.
Combin. Comput., 9(1991), 57-63.
[4]. E. J. Cockayne, Domination of undirected graphs - A survey. In theory and Applications of Graphs. LNM 642, Springer - Verlag,
(1978), 141-147.
[5]. E. J. Cockayne and S. T. Hedetniemi, Towards a theory of domination in graphs, Networks, 7, (1977), 247-261.
[6]. F. Harary, Graph theory, Addison Wesley, Reading Mass (1969).

Abstract: In this paper we have to study on the problem of 'Corruption' in different ways by using mathematical modelling. Also, we have to try a study of discrete model of corruption in the difference equation form. That is the comparatively mathematical study between the discrete model of corruption in the difference equation form and mathematical corruption model in the exponential form.
The problem of corruption is everywhere, so we will try to find the solution for the problem of corruption in the society. Therefore, how to measure the corruption in the society of any field or any country in the world? So, we have found the formula that is Mathematical corruption model for measuring the corruption in the society of any field or any country of the world. When we measure the corruption in the society then there will be no difficult to remove the corruption from the society of any country in the world.
Keywords: mathematical thinking, corruption mentality, modelling, applied.

[1]. Andriole SJ (1983) Handbook of Problem Solving: An Analytic Methodology Princeton, NJ: Petrocelli Books.
[2]. Blum, W. Niss, M.(1991). Applied Mathematical Problem Solving, Modeling, Applications and links to other subjects- state, trends and issues in mathematics instruction. In: Educational studies in Mathematics, 22(1), 37-68.
[3]. BhaskarDasgupta, Applied Mathematical Methods published by Darling Kindersley (India) Pvt. Ltd. Delhi.
[4]. Bender EA (1978) An Introduction to Mathematical Modeling New York: John Wiley andSons.
[5]. Daniel A. Murray, Introductory Course in Differential equations, Orient Longman ltd.Harlow and London (1993).
[6]. De Lange, J. (1996). Real Problems with Real world Mathematics. In L. Alsina & al.(Eds.) proceeding of the 8th Int. Congress on Math.1 Education (pp. 83-110). Seville: Thales.

Abstract: In this paper we introduce the concept of ternary  - semi group and we proved some properties of prime  -ideals and fuzzy weakly completely prime  -ideals of ternary  - semi groups. It is proved that, in a ternary  - semi group S if A S , then the following statements are equivalent (1) A is prime -ideal of a ternary  - semi group S (2) The characteristic function A C of A is a fuzzy weakly completely prime  - ideal of S .
Key words:  - semi group, Ternary  - semi group, prime  -ideals, fuzzy weakly completely prime ideals.

[1]. F. M. Sioson, "Ideal theory in ternary semi groups", Math.Japon. 10 (1965) 63–84.
[2]. J. Los, "On extending of models I", Fund.Math. V. 42 (1955) pp. 512-517.
[3]. Kim. J. , "Some fuzzy semi prime ideals in semi groups", Journal of Chungcheong Mathematical Society, 3(22),(2009); 277-288.
[4]. Lehmer.D.H. "A Ternary Analogue of Abelian Groups", Amer. Jr. of Maths. 59 (1932), 329-338.
[5]. Lyapin.E.S. "Realization of ternary semi groups", (Russian) Modern Algebra pp.43- 48(Leningrad Univ.), Leningrad (1981).
[6]. M. K. Sen, "On  -semi group", Proc. of InternationalConference on Algebra and its Applications.Decker Publication, 1981,
New York 301.

Abstract: In this paper, we introduce the notion of Generalized contractive type mappings in complex valued metric space and establish fixed point theorem for these mappings.
Keywords: Complex valued metric space, Generalized contractive maps,fixed point.

[1]. Azam, A. Fisher, and Khan, M. Common fixed point theorems in complex valued metric space, Numerical Functional Analysis and Optimization,vol.32,pp.243-253,2011.
[2]. Banach, S. Sur les operations dans les ensembles etleur application aux equations integrals ,Fundamenta Mathematicae,vol.3,pp.133-181,1992.
[3]. Rouzkard, F. And Imdad, M. Some common fixed point theorems on complex valued metric space, Computers & Mathematics with Applications,vol.64, no.6,pp.1866-1874,2012.
[4]. Verma ,R.K. and Pathak, H.K. Some common fixed point theorems using property(E.A) in complex valued metric space, Thai journal of Mathematics,2012.
[5]. Karapınar, E, Samet, B: Generalized α-ψ contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal.2012, Article ID 793486 (2012). doi:10.1155/2012/793486.

Abstract: Non-parametric and semi-parametric Bayesian regression is useful tools for practical data analysis. They provide posterior mean or median estimates, confidence bands and estimates of other functional without having to rely on approximate normality of estimators. The data that are analyzed are the rice yield in a district of Andhra Pradesh state and also the state data. We use a Bayesian threshold model with non-linear functions of four relevant continuous variables considered for the study. Inferences are based on Markov Chain Monte Carlo technique. The dependence of the results on the hyper parameters of the estimated variance components are analyzed. The effects of the covariates on rice yield in the district are compared with those of the state data.

[1]. Augustine, N.H. et.al. (2007) A spatial model for the needle losses of pine-trees in the forests of Baden-Wurttemberg: an application of Bayesian structured additive regression. Appl. Statist.56,29-50.
[2]. Congdon, P.(2006) A model for non-parametric spatially varying regression coefficients Computnl. Statist. Data Anal. , 50,422-455.
[3]. Eilers, P. and Marx, B.(1996) Flexible smoothing using B-splines and penalizes likelihood. Statist. Sci., 11,89-121.
[4]. Fahrmeir, L. and Tutz, G. (2001) Multivariate Statistical Modeling based on Generalized Linear Models. New York: Springer.