iosr-Jm   Volume-11 ~ Issue-5 ~ Sep-Oct 2015

Abstract: Linear Programming Problems have been solved using traditional Algorithms of Simplex Methods. In this paper a modified approach to Simplex Method has been worked upon. This approach involves a determinant at every step to perform the next iteration till the optimum solution is obtained.

Keywords: Constraints, Determinant, Feasible Solution, Linear Programming Problem, optimum Solution, Pivot, Simplex Method , Slack Variables, Surplus Variables.

[1]. Klee V, Minty G.J, How Good is the Simplex Algorithm? In: Shisha O, editor Inequalities III, New York Academic Press, 1972, 158-172.
[2]. Stojkovic N.V, Stanimirovic P.S, Two Direct Methods in Linear Programming, European Journal of Operations Research, 2001: 417-439.
[3]. Arsham H, Damage T, Grael J, An Algorithm for Simplex Tableau Reductions; The Push to Pull Solutions Strategy, Applied Mathematics and Computation, 2003, 525-547.
[4]. Prem Kumar Gupta, D S Hira, Operations Research (S. Chand and Co Ltd., 2008).
[5]. Taha Hamdy. A, Operations Research, (Dorling Kindersley, India Pvt. Ltd., Licences of Pearson Education in South Asia)
[6]. Helcio Vieira Junior, Marcos Pereira, Estellita Lins, An Imrpved Initial Basis for Simplex Algorithm Computers and Operations Research 32 (2009). 1984-1993.

Abstract:In this paper, we introduce the concepts of pure po-ternary Γ-ideal, weakly pure po-ternary Γ-ideal and purely prime po-ternary Γ-ideal in an ordered ternary Γ-semiring. We obtain some characterizations of pure po-ternary Γ-ideals and prove that the set of all purely prime po-ternary Γ-ideals is topologized.
Keywords: ternary Γ-semiring; ordered ternary Γ-semiring; weakly regular; pure po-ternaryΓ-ideal; weakly pure po-ternary Γ-ideal; purely prime po-ternary Γ-ideal; topology.

[1]. Ahsan. J, Takahashi. M, Pure spectrum of a monoid with zero, Kobe J. Mathematics 6 (1989) 163–182.
[2]. Bashir. S, Shabir. M, Pure ideals in ternary semigroups, Quasigroups and Related Systems 17 (2009) 149–160.
[3]. Changphas. T, Bi-ideals in ordered ternary semigroups, Applied Mathematical Sciences 6 (2012) 5943–5946.
[4]. Changphas. T, Bi-ideals in ternary semigroups, Applied Mathematical Sciences 6 (2012) 5939–5942.
[5]. Chinram. R, Saelee. S, Fuzzy ideals and fuzzy filters of ordered ternary semigroups, Journal of Mathematics Research 2 (2010) 93–97.
[6]. Dixit. V. N, Dewan. S, A note on quasi and bi-ideals in ternary semigroups, Internat. J. Math. Math. Sci. 18 (3) (1995) 501–508.
[7]. Iampan. I, On ordered ideal extensions of ordered ternary semigroups, Lobachevskii J. Math. 31 (1) (2010) 13–17.

Abstract:In this article, a new method of analysis for linear and nonlinear stiff problems using the Adomian Decomposition Method (ADM) is presented. To illustrate the effectiveness of the ADM an example from linear and nonlinear stiff problems have been considered and compared with the single-term Haar Wavelet series (STHW)[9] and with exact solutions of the problems, and are found to be very accurate. Error graphs for the linear and nonlinear stiff problems are presented in a graphical form to show the efficiency of this ADM. This ADM can be easily implemented in a digital computer and the solution can be obtained for any length of time.
Keywords: Linear stiff differential equations, nonlinear stiff differential equations, Single-term Haar wavelet series, Adomian Decomposition Method.

[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] C. F. Curtiss and J. O. Hirschfelder, Integration of Stiff Equations, Proc. Nat. Acad. Sci., 38(1952), 235-243.
[4] 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.
[5] 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.

Abstract:The aim of this paper, is to use a more realistic model which incorporates the effects of Brownian motion and thermophoresis for studying the effect of a uniform heat source on the onset of convective instability in a confined medium filled of a Newtonian nanofluid layer and heated from below, this layer is assumed to have a low concentration of nanoparticles. The linear study in the rigid - rigid case which was achieved in this investigation shows that the thermal stability of Newtonian nanofluids depends of the volumetric heat delivered by the internal source, the Brownian motion, the thermophoresis and other thermos-physical properties of nanoparticles. Our problem will be solved using a technique of converting a boundary value problem to an initial value problem, in this technique we will also approach the searched solutions with polynomials of high degree.
Keywords: Linear stability, Nanofluid, Brownian motion, Thermophoresis, Internal heat source, Power series, Realistic approach, Rigid-Rigid case.

1] S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: D.A.Siginer, H.P. Wang (Eds.), development and applications of non-newtonian flows, ASME FED-231/MD, 66, 1995, 99-105.
[2] J. Buongiorno, Convective transport in nanofluids, Journal of Heat Transfer, 128, 2006, 240-250.
[3] D.Y. Tzou, Thermal instability of nanofluids in natural convection, International Journal of Heat and Mass Transfer, 51 2008, 2967-2979.
[4] D.Y. Tzou, Instability of nanofluids in natural convection,ASME Journal of Heat Transfer, 130, 2008, 372-401.
[5] D.A. Nield and A.V. Kuznetsov, The onset of convection in a horizontal nanofluid layer of finite depth, European Journal of Mechanics B/Fluids, 29, 2010, 217-233.

Abstract:Let P𝑘+1 denote a path of length 𝑘 and let C𝑘 denote a cycle of length 𝑘. As usual 𝛫𝑛 denotes the complete graph on 𝑛 vertices. In this paper we investigate decompositions of line graph of 𝛫𝑛 into p copies of P5 and q copies of C4 for all possible values of p ≥ 0 and q ≥ 0.

Keywords: Path, Cycle, Graph Decomposition, Complete graph, Line graph.

1] A.A. Abueida, M.Daven Multidesigns for graph-pairs of order 4 and 5 Graphs combin. 19 2003 433-447
[2] A.A. Abueida, M.Daven Multidecomposition of the complete graph Ars combin.72 2004 17-22
[3] A.A. Abueida, M. Daven and K.J. Roblee Multidesigns of the λ–fold complete graph-pairs of orders 4 and 5 Australas.J. Combin. 32 2005 125- 136
[4] A.A. Abueida, T. O'Neil Multidecomposition of K𝑚(λ) into small cycles and claws Bull. Inst. Comb. Appl. 49 2007 32-40
[5] Discuss. Math. Graph Theory. 34 2014 113-125
[6] B. Alspach Research Problems, Problem 3 Discret Math 36 1981 333.

Abstract: A new algebraic structure Great Algebra or simply Gr-Algebra with three binary operations is defined. Additive Commutative Gr-Algebra, Multiplicative Commutative Gr-Algebra, Bi- Commutative Gr-Algebra, Tri-commutative Gr-Algebra, Gr-Algebra with multiplicative identity, Multiplicative Gr-Algebra Unit, Division Gr-Algebra, Field Gr-Algebra, R-identity and L-identity and identity of a Gr-Algebra are defined.

Keywords : Gr-Algebra, Division Gr-Algebra, Field Gr-Algebra, R-identity and L-identity and identity, Right-Unit, Left-Unit.

[1]. I.N.Herstein, Topics In Algebra, Wiley Eastern Limited.
[2]. John T.Moore, The University Of Florida /The University Of Western Ontario, Elements Of Abstract Algebra, Second Edition, The Macmillan Company, Collier-Macmillan Limited, London,1967.
[3]. K.Muthukumaran And M.Kamaraj, "Artex Spaces Over Bi-Monoids", "Research Journal Of Pure Algebra", 2(5), May 2012, Pages 135-140.
[4]. K.Muthukumaran And M.Kamaraj, "Subartex Spaces Of An Artex Space Over A Bi-Monoid" ,"Mathematical Theory And Modeling"(With Ic Impact Factor 5.53), An Usa Journal Of "International Institute For Science, Technology And Education", Vol.2, No.7, 2012, Pages 39 – 48.
[5]. K.Muthukumaran And M.Kamaraj, "Bounded Artex Spaces Over Bi-Monoids And Artex Space Homomorphisms", "Research Journal Of Pure Algebra", 2(7), July 2012, Pages 206 – 216.
[6]. K.Muthukumaran And M.Kamaraj, "Some Special Artex Spaces Over Bi-Monoids", "Mathematical Theory And Modeling"(With Ic Impact Factor 5.53), An Usa Journal Of "International Institute For Science, Technology And Education", Vol.2, No.7, 2012, Pages 62 – 73.

Abstract:Let G (V, E) be a simple, finite, undirected connected graph. A non – empty set S  V of a graph G is a dominating set, if every vertex in V – S is adjacent to atleast one vertex in S. A dominating set S  V is called a locating dominating set, if for any two vertices v, w  V – S, N(v)  S  N(w)  S. A locating dominating set S  V is called a co – isolated locating dominating set, if there exists atleast one isolated vertex in <V – S >. The co – isolated locating domination number  cild is the minimum cardinality of a co – isolated locating dominating set. In this paper, the number cild is obtained for unicyclic graphs.
Keywords: Dominating set, locating dominating set, co – isolated locating dominating set, co – isolated locating domination number.

[1] Ore. O., Theory of Graphs, Amer. Math. Soc. Coel. Publ. 38, Providence, RI, 1962.
[2] Rall. D. F., Slater. P. J., On location domination number for certain classes of graphs, Congrences Numerantium, 45 (1984), 77 – 106.
[3] S.Muthammai., N.Meenal., Co - isolated Locating Domination Number for some standard Graphs, National conference on Applications of Mathematics & Computer Science (NCAMCS-2012), S.D.N.B Vaishnav College for Women(Autonomous), Chennai, February 10, 2012, p. 60 – 61.
[4] S.Muthammai., N.Meenal., Co - isolated Locating Domination Number of a Graph, Proceedings of the UGC sponsored National Seminar on Applications in Graph Theory, Seethalakshmi Ramaswamy College (Autonomous), Tiruchirappalli, 18th& 19th December 2012, p. 7 – 9.
[5] Muthammai.S., Meenal. N., Co - isolated Locating Domination Number for Cartesian Product Of Two Graphs, International Journal of Engineering Science, Advanced Computing and Bio - Technology, VOL 6, No.1, January – March 2015, p. 17 – 27
[6] Muthammai.S., Meenal. N., The Number of Minimum Co – isolated Locating Dominating Sets Of Cycles, International Research Journal of Pure Algebra – 5(4),April - 2015, p. 45 - 49

Abstract:Time-delay dynamical systems are utilized to describe many phenomena of physical interest. Special effort of this paper is given to Hopf bifurcation of time-delay dynamical systems. Linear analysis is carried out to determine the critical conditions under which Hopf bifurcation can occur. Delays being inherent in any biological system, we seek to analyze the effect of delays on the growth of populations governed by the logistic equation. The local stability analysis and local bifurcation analysis of the logistic equation with two delays is carried out considering one of the delays as a bifurcation parameter.
Keywords: Logistic equation, delay, stability, Hopf bifurcation.

[1] E. H. Ait Dads and K. Ezzinbi (2002), Boundedness and almost periodicity for some state- dependent delay differential equations,
Electron. J. Differential Equations 2002, No. 67, pp.1-13.
[2] W. C. Allee (1931), Animal Aggregations: A Study in General Sociology, Chicago University Press, Chicago.
[3] O. Arino and M. Kimmel (1986), Stability analysis of models of cell production systems, Math. Modelling 7, 1269-1300.
[4] S. H. Strogatz, Nonlinear Dynamics and Chaos. Perseus Books, 1994.
[5] C. Sun, M. Han and Y. Lin, "Analysis of stability and Hopf bifurcation for a delayed logistic equation," Chaos, Solitons and
Fractals, vol. 31, pp. 672-682, 2007.
[6] R. L. Kitching , Time, resources and population dynamics in insects, Australian Journal of Ecology vol. 2, pp. 31-42, 1997.
[7] M. M. Rao and Preetish K. L., Stability and Hopf Bifurcation Analysis of the Delay Logistic Equation, Cornell university library,
arXiv: 1211.7022v1 [q-bio.PE] 29 Nov 2012.

Abstract:Every linear transformation on a finite dimensional vector space gives different matrix representation w.r.t different basis. But as they are representation of same linear transformation. Therefore all the matrix representation of the transformation must have common properties of the transformation and must have a canonical representation. Here it is found out such canonical form when the field is algebraically closed. If eign vectors of the transformation can span the vector space, then canonical form will be diagonal matrix with eign values are diagonal elements. But if they not span the vector space, then using two famous well-known theorems (1) Primary Decomposition theorem and (2) Cyclic Decomposition theorem we get the famous Jordan Canonical Form which is the simplest representation of the linear transformation for algebraically closed field.

Keywords– Linear transformation, Primary Decomposition, Cyclic decomposition, Jordan Canonical form.

[1]. kenneth hoffman & ray kunze, linear algebra, chapter 6 &7

Abstract:In coastline, the boundary-fitted curvilinear grids are appropriate to make the grid model and build up the simple boundary conditions, which are more mathematic. The method of lines (MOL) is a general procedure for the solution of time dependent partial differential equations (PDEs. The methods of lines (MOL) are more feasible and comparable to the regular finite difference method in terms of computational time, accuracy and numerical stability. Thus, we have tried to present the new numerical solution technique to simulate the 2004 Indonesian tsunami. At first, this numerical solution technique set up on a curvilinear grid model where the vertically integrated shallow water equations are solved by using MOL. The boundary fitted curvilinear grids are presented along the coastal and island boundaries.

[1] Cash JR (2005). Efficient time integrators in the numerical method of lines. J. Computational and Appl. Mathematics, vol. 183, 259-274.

[2] G. D. Roy, A. B. M. Humayun Kabir, M. M. Mondol, Z. Haque: "Polar coordinates shallow water storm surge model for the coast of Bangladesh" Dyn. Atms. Oceans., 29, pp 397 – 413. (1999).

[3] G. D. Roy: "Inclusion of off-shore islands in a transformed coordinates shallow water model along the coast of Bangladesh" Environment International, 25 (1), pp 67- 74. (1999)

[4] Ismail AIM, Karim MF, Roy GD, Meah MA (2007). Numerical Modelling of Tsunami via the Method of lines. Int. J. Mathematical, Physical and Engr Sci., 1(4), 213 - 221.

[5] Johns B, Dube SK, Mohanti UC, Sinha PC (1981). Numerical Simulation of surge generated by the 1977 Andhra cyclone.Quart. J. Roy. Soc. London 107, 919 – 934.

Abstract:In this paper an interesting and famous realistic electronic circuit problem is discussed using the Adomian Decomposition Method (ADM) is presented. To illustrate the effectiveness of the ADM an example from linear time-varying electrical circuit problem have been considered and compared with the Runge-Kutta Butcher algorithm (RKB)[5] and with exact solutions of the problems, and are found to be very accurate. Error graphs for the linear time-varying electrical circuit problem are presented in a graphical form to show the efficiency of this ADM. This ADM can be easily implemented in a digital computer 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] K. Balachandran and K. Murugesan, "Analysis of Electronic Circuits Using the Single-Term Walsh Series Approach," Int. J. Of
Electronics, vol. 69, no. 3, pp. 327-332, 1990.
[3] L.O. Chua and P.M. Lin, "Computer-Aided Analysis of Electronic Circuits", Prentice-Hall, New Jersey, USA, 1975.
[4] J. C. Butcher, "The Numerical Methods for Ordinary Differential Equations", John Wiley & Sons, U.K., 2003.
[5] K. Murugesan, N.P. Gopalan, and Devarajan Gopal," Error Free Butcher Algorithms for Linear Electrical Circuits", ETRI Journal,
vol. 27, no. 2, April 2005, pp. 195-205.
[6] 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.

Abstract: In this paper we fit a time series model that best describes the rainfall pattern of Uasin Gishu county from the general ARIMA family and generate the values (p,d,q)(P,D,Q)s. The model that best fitted the Kapsoya historical rainfall data was SARIMA (0,0,0)(,0,1,2)12. This model is used to forecast average expected monthly rainfall statistics for two years. For verification and data fitting to the model, R computer software was employed. The data used is real rainfall data from Kapsoya meteorological station in Uasin Gishu County.

Keywords: Time series, Forecasting, Rainfall values, ARMA, ARIMA, SARIMA.

[1]. Box,G. & G.Jenkins, (1976). Time series analysis: forecasting and control. (Revised ed) Holden day, San Francisco.
[2]. Brockwell,P.J. & Davis, R.A. (1996). Introduction to Time series and forecasting. (2nd ed.). New York: Springer.
[3]. Chatfield, C.,(1991). The Analysis of Time Series. An Introduction, (4th ed)., Chapman and Hall: London.
[4]. Chatfield,C. (1996). The analysis of time series- an introduction,(5th ed) ,Chapman and Hall, U.K.
[5]. Cromwell, J.B., Labys, W.C and Terraza, M., (1994). Univariate Tests for Time Series Models. A Sage Publication, (7-99)96., London.
[6]. Diebold, F.X., Kilian, L. and Nerlove, M., (2006). Time Series Analysis. Working Paper No.06-011, University of Maryland, College Park.
[7]. Enders, W., (1995). Applied Econometric Time Series, John Wiley and Sons, Inc., New York.

Abstract: Rayleigh-Bénard-Marangoni convection in a relatively hotter or cooler layer of liquid is studied theoretically by means of modified linear stability theory. The upper surface of the layer is considered to be non-deformable free where surface tension gradients arise on account of variation of temperature and the lower boundary surface is rigid, each subject to constant heat flux condition. The Galerkin technique is used to obtain the eigenvalue equation analytically. This analysis predicts that the onset of convection in a relatively hotter layer of liquid is more stable than a cooler one under identical conditions, irrespective of whether the two mechanisms causing instability act individually or simultaneously, and that the coupling between the two agencies causing instability remains perfect.
Keywords: Buoyancy, Convection, Coupling, Linear stability, Surface tension, Insulating.

[1]. M.J. Block, Surface tension as the cause of Bénard cells and surface deformation a liquid film, Nature 178, 1956, 650–51.
[2]. J.R.A. Pearson, On convection cells induced by surface tension, J. Fluid Mechanics, 4, 1958, 489-500.
[3]. H. Bénard, Les tourbillons cellulaires dans une napple liquide, Rev. gén. Sciences pures at appl. 11, 1900, 1261–1271.
[4]. H. Bénard, Les tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en régime permanent, Ann.
Chimie (Paris) 23, 1901, 62–144.
[5]. L. Rayleigh, On convection currents in a horizontal layer of fluid, when the higher temperature is underside, Phil. Mag. 32, 1916,
529-546.
[6]. D.A. Nield, Surface tension and buoyancy effects in cellular convection. J. Fluid Mechanics, 19,1964, 571-574.