iosr-Jm   Volume-5 ~ Issue-6

Abstract: The aim of this paper is to prove a common fixed point theorem which generalizes the result of Brian Fisher [1] and etal. by weaker conditions. The conditions of continuity, compatibility and completeness of a metric space are replaced by weaker conditions such as reciprocally continuous and compatible, weakly compatible, and the associated sequence.

Keywords: Fixed point, self maps, reciprocally continuous, compatible maps, weakly compatible mappings.

[1] B.Fisher and etal, "Common Fixed Point Theorems for compatible mappings", Internat.J. Math. & Math. Sci, 3(1996), 451-456.

[2] G .Jungck, "Compatible Mappings and Common Fixed Points", Inst. J. Math. Math. Sci .9(1986), 771-779.

[3] G.Jungck, B.E. Rhoades, Fixed Point for set valued Functions without Continuity, Indian.J. Pure.Appl.Math,3(1998), 227-238.

[4] G. Jungck., B.E. Rhoades, Fixed point for set valued functions without continuity, Indian J. Pure.Appl.Math, 3 (1998), 227-238.

[5] V.Srinivas, R.Umamaheshwar Rao ,A Fixed point Theorem for Weekly Compatible Mappings , Journal of Mathematical Sciences & Engineering Applications,1(2007),41-48.

[6] V.Srinivas, R.Umamaheshwar Rao, Common Fixed Point Theorem for Four Self Maps, International Journal of Mathematics Research, 2(2011 ),113-118.

Abstract: In this paper we discussed the stability of the null solution of the second order differential equation . Under some unusual assumptions we obtain new stability results for this classical equation.

[1] Bellman.R.,Staility Theory of Differential Equations,McGraw-Hill,New York
[2] Burton T.A, and Furumochi,Tetsuo (2001) Fixed points and prolem in stability Theory for ordinary and functional Differential quations, Dynamical System. April 10, 89-116
[3] Burton T.A, and Furumochi,Tetsuo (2002) Asymptotic behaviour of solution of functional Differential Equations, by fixed point theorem Dynamical system 11,499-519.
[4] Burton T.A, and Furumochi,Testuo (2002) Krasnoselskiis fixed point theorem and stability of nonlinear Analysis 49, 445-454.
[5] Burton T.A, and Furumochi,Testuo (2005) Asymptotic behaviour of non linear functional Differential Equations, by Schauder's theorem Nonlinear Stud. 12,73-84.
[6] Coddington, E.A. and Levinson, Theory of Differential Equations,McGraw-Hill,New York.
[7] C. Corduneanu, Principles of Differential and Integral Equations, Allyn and Bacon, Boston, 1971.
[8] Gh. Moro¸sanu, C. Vladimirescu, Stability for a nonlinear second order ODE, CEU Preprint, August 2003, to appear in Funkcialaj Ekvacioj.
[9] Hale, Jack K., Ordinary Differential Equations, Wiley, New York, 1969.
[10] Hartman, Philip, Ordinary Differential Equations, Wiley, New York, 1964.

Abstract:The field equations for perfect fluid coupled with mass less scalar field are solved with conditions 𝑝=𝜌 𝑎𝑛𝑑 𝑅=𝑘𝐴𝑛 (where k and n both are constant.) for five dimensional space-time in General Theory of Relativity. Also various physical and geometrical properties of the model have been discussed. Keywords - Five dimensional space-time, mass less scalar field, perfect fluid.

[[1] Gasperini, M.: Phys. Rep. 373, 1(2003)

[2] Gron, P.: Math. Phys 27, 1490 (1986)

[3] Krori, K. D., Goswami, A. K., Purkayastha, A. D.: J. Math. Phys. 36, 1347 (1995)

[4] Li, X. Z., Hao, J. G.: Phys. Rev. D. 68, 083512 (2003)

[5] Lorenz, D.: Astrophys.Space Sci.98, 101(1984)

[6] Lorenz, D.: J. Phys. A: Math. Gen. 16, 575 (1983)

[7] Matravers, D. R.: Gen. Relativ. Gravit. 20, 279 (1988)

[8] Mohanty, G., Tiwari, R. N. Rao, J. R.: Int.J. Theor. Phys. 21, 105 (1982)

[9] Mohanty. G., Mahanta K.L. and .Bishi B.K.,Astrophys Space Sci., 310(2007)273.

[10] Mohanty. G.,Mahanta K.L. and Sahoo R.R.,Astrophys.Space Sci., 306(2006)269

Abstract: The present paper is devoted to the study of the compositions of operator of generalized function 𝑮𝝆,𝜼,𝜸 𝒂,𝒛 defined in 1 and their applications. The compositions with Riemann-Liouville fractional integral and differential operator are also derived. As applications of our main-results some known results for generalized Mittag-Leffler function due to Kilbas et al. 2 𝑎𝑟𝑒 𝑐𝑖𝑡𝑒𝑑. The results involving the R-function 3 are also obtained as special cases of our main findings.

Key words: Generalized function, R-function, Generalized Mittag-Leffler function, Riemann-Liouville fractional calculus, Generalized fractional integral operators.

[1]. Harish Nagar and Anil Kumar Menaria, "On Generalized Function Gρ,η,γ a,z And It's Fractional Calculus" Vol 4, SPACE , ISSN 0976-2175
[2]. Kilbas, A.A., Saigo, M. and Saxena, R, K. ;( 2004): Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral transform and Special functions Vol.15, No.1, 31-49.
[3]. Lorenzo, C.F. and Hartley, T.T. ;( 2000): R-function relationships for application in the fractional calculus, NASA, Tech, Pub.210361, 1-22.
[4]. Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G.; (1953): Higher Transcendental Functions, Vol.-I, McGraw-Hill, New York.
[5]. Lorenzo, C.F. and Hartley, T.T. ;( 1999): Generalized functions for the fractional calculus, NASA, Tech, Pub.209424, 1-17.
[6]. Lorenzo, C.F. and Hartley, T.T. ;( 2000): Initialized fractional calculus, International J. Appl.Math.3, 249-265.

Abstract: Categorical data are often stratified into two dimensional I × J tables in order to test the independence of attributes. Agresti (1999) has discussed testing methods with partitioning property of chi square to extract components that describe certain aspects of the overall association in a table. In this work an attempt has been made to study the pattern in which sub-tables exhibit a sign of reverse association when compared to a significant association of attributes with regard to the original I × J table. Simulation studies and subsequent results of vote counting method indicate that 2 × 2 tables have the reversal component association when compared to higher order sub-tables; interestingly, in more than 90% cases. The computationally extensive and exhaustive procedure provide a better tool to understand the association between the categories that focus on the strongest differences among all comparisons, and could be practically and other aspects in experimental studies such as clinical trials.

Keywords: Contingency tables, Chi-Square test, Component Analysis.

[1] J.Berkson,. Some difficulties of interpretation encountered in the application of the chi-square test, Journal of the American
Statistical Association 33, 1938, 526-536.
[2] H.W,Norton, Calculation of chi-square for complex contingency tables, J. Amer. Statist. Assoc. 40, 1945, 251-258.
[3] E.H.Simpson, The interpretation of interaction in contingency tables, J. Roy. Statist. Soc. B 13,1951, 238-241.
[4] W.G. Cochran, Some methods for strengthening the common chi-square tests, Biometrics 24, 1954, 315-327.
[5] N. Mantel, and W.Haenszel, Statistical aspects of the analysis of data from retrospective studies of disease, J. Nat. Cancer Inst. 22,
1959, 719-748.
[6] L.A. Goodman, Simple methods for analyzing three-factor interaction in contingency tables, J Amer. Statist. Assoc. 59, 1964, 319-
352.
[7] L.A. Goodman, On partitioning 2  and detecting partial association in three-way contingency tables, J. Roy. Statist. Soc. B 31,
1969, 486-498.
[8] C.R. Blyth, On Simpson's paradox and the sure thing principle. J Amer. Statist. Assoc. 67, 1972, 364-366.
[9] A.Agresti, Categorical Data Analysis, (New York: Wiley & Sons 1990) pp 51-54
[10] A.Agresti, Exact inference for categorical data: recent advances and continuing controversies, Statistical Methods and Applications,
20, 1999, 2709-2722.

Paper Type	:	Research Paper
Title	:	Backlund's Theorem for Spacelike Surfacesin Minkowski 3-Space
Country	:	Saudi Arabia.
Authors	:	M. T. Aldossary
	:	10.9790/5728-0562430

Abstract: By using the method of moving frames the Backlund's theorem and its applicationfor spacelike surfaces is introduced. The results leads to correspondence between the solutions of the Sine-Gordon equation and spacelike surfaces of constant positive Gaussian curvatures.

Key Words: Line Congruence; Backlund's Theorem; Sine-Gordon Equation.

[1] Abdel-Baky, R. A.;The Backlund's Theorem in Minkowski 3-Space, AMC 160, pp. 41-50, (2005).

[2] Chern, S. S. and Terng, C. L.;An analogue of Backlund's Theorem in Affine Geometry, Rocky Mountain J. of Math.,V. 10. N.1, (1980).

[3] Chern, S. S.;Geometrical interpretation of Sinh-Gordon equation, Ann. Polon. Math. 39, pp74-80, (1980).

[4] Chen, W.H.;Some results on spacelike surfaces in Minkowski 3-Space, Acta. Math.Sincia, 37, 309-316, (1994)(In Chinese).

[5] Eisenhart, L. P.;A Treatise in Differential Geometry of curves and surfaces, New York, Ginn Camp., (1969).

[6] Kobayashi, S. and Nomizu, K.;Foundations of Differential Geometry, I, II New York, (1963), (1969).

[7] Mc-Nertney, L. V.;One-Parameter families of surfaces with constant curvature in Lorentz three-space, Ph.D. Thesis, Brown University, (1980).

[8] Tian, C.;Backlund transformations on surfaces with in , J. Geom. Phys. 22, pp.212-218, (1997).

[9] Zait, R. A.;Backlund transformations and solutions for some Evolution equations, PhysicaScripta, Vol. 57, pp.545-548, (1998).

Abstract: This research computes the optimal control and state of the one-dimentional energized wave equation using the Finite Element Technique ( FET). The paper has to do with all the vital computational elements as in the derivation of the finite element algorithm . With these recalls, various numerical optimal controls and states were considered at various levels of discretization.

Keywords: Differential Equation, Energized Wave Equation, Finite Element Technique, Optimal Control, Optimal State.

[1] M. D. Raisinghania, Advanced differential equations,S. Chand and Company Ltd, New Delhi, 2010.
[2] S. A. Reju, M. A. Ibiejugba, and J. D. Evans, Optimal control of the wave propagation problem with the extended conjugate
gradient method, Intern. J. Computer Math. 77(3),2001,425-439.
[3] M.Bawa, Application of finite element technique to the optimal control of wave equation with energy effect, unpublished M.Tech
thesis, Federal University of Technology, Minna, Nigeria,2001.
[4] H. J.Pain, The physics of vibrations and waves, 2nd edition, John Wiley and Sons, 1979.
[5] M. G. Singh, and A. Titli, Systems (Decomposition, Optimization and Control) ,Pergamon Press,1978.
[6] P. Duchateau, and D. W. Zachmann, Partial differential equations, McGraw-Hill Publishing Company,1986.
[7] S. S. Rao, The finite element method in engineering, Pergamon Press, 1989.

Abstract:A logistics network consists of suppliers, manufacturing centers, warehouses, distribution centers, and retail outlets as well as channels for the flow of raw materials, work-in-process inventory, and finished products between the facilities . In many cases, the problems of each logistics functions are treated as isolated functions. There are: network design; information flow; transportation; inventory; and warehousing, material handling, and packaging. Whereas, the basis of performance improvement in integrated logistics network is total cost analysis. That is minimizing the total cost of transportation, warehousing, order processing and information, lot quantity, and inventory carrying cost. In this paper, We consider an integrated logistics problem in minimizing the total logistics cost and provide model for logistics network optimization in form of a mathematical programming model. An integrated model of logistics network minimizes total physical distribution costs by simultaneously determining optimal plants and warehouse locations, flows in the resulting network, shipment compositions and shipment frequencies in the network using heuristics methodology. The objective is to optimize the production and distribution plan so as to minimize its total logistics cost.

Keywords: Transportation Sciences, Integrated Model, Mathematical Programming, Network Optimization.

[1]. Ambrosino, D., Scutella, M.G., 2005, "Distribution network design: New problems and related models", European Journal of
Operational Research, Vol. 165 No. 3, pp. 610–624
[2]. Beamon, B.M., 1998, "Supply Chain Design and Analysis: Models and Methods", International Journal of Production Economics,
Vol. 55 No. 3, pp. 281-294.
[3]. Beamon, B.M., 1999, "Measuring Supply Chain Performance", International Journal of Operations and Production Management,
Vol. 19 No. 3, pp. 275-292.
[4]. Blanchard, B.S., Logistics Engineering And Management, 5th Edition Prentice-Hall, Inc., Upper Saddle River, NJ, 1998.
[5]. Blumenfeld, D.E., Burns, L.D., Daganzo, C.F., Frick, M.C., and Hall, R.W., 1987, "Reducing logistics costs at General Motors",
Interfaces , Vol. 17 No. 1, pp. 26-47
[6]. Bowersox,D.J., Closs, D.J., Logistical Management: The Integrated Supply Chain Process, 1st edition, The McGraw-Hill
Companies, 1996.
[7]. Bramel, J., Simchi-Levi, D, The logic of logistics, Springer Series in Operation Research, Springer, 1997.

Abstract: Control theory has developed rapidly over the past few decades and it is now established as an important area of contemporary applied mathematics. Optimal control problem is a mathematical programming problem involving a number of stages where each stage evolves from the previous stage in a prescribed manner. In this paper, we are concerned with optimal control of delay differential equations whose costs functional are quadratic and whose state variables are governed by linear delay differential equations. We now used the multiplier method in solving the resulting problem.

Keywords: Bio-Mathematics, Delay Differential Equations, Multiplier Method, Optimal Control

[1]. Martin, A and Ruan, S. Predator – Prey Models with Delay and Prey Harvesting, Journal of Mathematical Biology, Vol. 43, 2001, 247-267.

[2]. Raghothama, A. and Narayanan, S. Periodic Response and Chaos in NonLinear Systems with Parametric Excitation and Time Delay, Journal of Nonlinear Dynamics, Vol. 27, 2002, 341 – 365.

[3]. Neves, K.W. and Feldstein, A. Characterization of Jump Discontinuities for State Dependent Delay Differential Equations, Journal of Mathematical Analysis and Applications, Vol. 5, 1976, 689 – 707.

[4]. Wright, E.M. A Functional Equation in the Heuristic Theory of Primes, The Mathematical Gazette, Vol. 45, 1961, 15 – 26.

[5]. Cunningham, W.J. A Non-linear Differential Difference Equation of Growth, National Academy of Science, U.S.A., Vol. 40, 1954, 709 – 713.

[6]. Volterar, V. Mathematical Theory of Hereditary Phenomenon, Journal of Mathematics Application, Vol. 7, 1928, 249 – 268.

[7]. Wangesky, P.J. and Canningham, W.J. Time Lag in Prey predator Population Models, Ecology Gazette, Vol. 38, 1957, 136 – 139. [8] Banks, H.T and Burns, J.A. Hereditary Control Problems using Numerical Methods based on Averaging Approximations, SIAM Journal of Control, Vol. 16, 1978, 169 – 190.

[9] Yorke, J.A. Non continuous Solution of Delay Differential Equations, Pro. Amer. Soc. Vol. 21, 1969, 648 – 652.

Abstract: Queuing theory has been applied to a variety of business situations related to customer involvement. The firm provides service facility and tries to keep the costs and time minimum to infuse goodwill among customers. This necessitates the study of service facility to obtain the number of customers and their waiting time. Control chart technique may be applied to analyze the services and the effective performance of concerns. Control chart constructed for the time spent in the system provides the prior idea about expected waiting time, maximum waiting time and minimum waiting time which guarantees customer's satisfaction. Keeping this in view, the construction of control chart for M /M /s queuing model with infinite capacity is proposed in this paper.

Keywords - waiting time, control limits, Poisson arrival and exponential service, s- servers

[1] Montgomery D.C., Introduction to statistical quality control (5th Edition John Wiley & Sons, Inc , 2005)

[2] Shore,H. General control charts for attributes, IIE transactions, 32, 2000, 1149-1160.

[3] Khaparde, M.V. and Dhabe, S.D. Control chart for random queue length N for (M/M/1):( ∞/FCFS) queueing model,International Journal of Agricultural and Statistical sciences, Vol.1, 2010, 319-334.

[4] Poongodi.T and Muthulakshmi. S. (2013), Control chart for waiting time in system of (M/M/1): (∞/FCFS) Queueing model, International Journal of Computer Applications, Vol 63, No.3, 2013, 1- 6.

[5] Gross, D. and Harris, C.M., Fundamentals of queueing theory ( 3rd edition, (1998), John Wiley and sons.

Abstract: This paper considered a two dimensional flow in x and y coordinate system and derived a Non- Homogenous Laplace Equation for groundwater flow using Darcy's equation which form the basis for the finite element discretization process. By discretizing the Hydraulic Conductivity and Piezometric Head in the x y direction, a flow direction vector of the groundwater system is obtained. Keywords –Finite element method, Hydraulic conductivity, Groundwater flow, Piezometric head, Laplace equation

[1] S. Jiban, R. K. Somashekar, K. L. Prakashand K. Shivanna, Investigation of heavymetals in crystalline aquifer groundwater from Different valleys of Bangalore, Karnataka, Journal of Geography andRegional Planning, 3(10), 2010, 262-270.

[2] J. Simunek and M. T. Van Genuchten, Contaminant transport in the unsaturated zone: Theory and modeling, in J. W. Delleur (Ed.), Thehandbook of groundwater engineering, 2nd ed., 22(New York: CRC Press, 2006).
[3] J. Simunek, Models of water flow and solute transport in the unsaturated zone, in M. G. Anderson and J. J. McDonnell (Eds.), Encyclopedia of hydrological sciences, 78 (Chichester, England: John Wiley & Sons, Ltd., 2005) 1171 – 1180.
[4] J. W. Delleur, Elementary groundwater flow and transport processes,in J. W. Delleur (Ed.), The handbook of groundwater engineering , 2nd ed., 3 (New York: CRC Press, 2006).

[5] A. Kamkar-Rouhani, 2D Modelling of groundwater flow using finite element method in an object-oriented approach, in R. Cidu& F. Frau (Eds.), IMWA symposium 2007: Water in mining environments, Cagliari, Italy, 2007

[6] P. I. Olasehinde, The groundwaters of Nigeria: A solution to sustainable national water needs, Inaugural Lecture Series 17, Federal University of Technology, Minna, Nigeria, 2010.

[7] C. P. Kumar, Groundwater Flow Models, in N. C. Ghosh& K. D. Sharma (Eds.), Groundwater modeling and management, (New Delhi: Capital Publishing Company, 2006) 153 – 178.

[8] T. J. Durbin, Groundwater flow and transport model, Report on Seaside groundwater basin, Monterey County, California, Timothy J. Durbin,Inc., Consulting hydrologists, CA,2007, 45- 48.

[9] A. Idris-Nda, Hydrogeophysical and hydrogeochemical characterization of the Bida Basin aquifer system,, doctoral thesis, FederalUniversity of Technology, Minna, Nigeria, 2010.

Abstract: In this paper, we were concerned with applying linear programming technique to determine optimum production of Usmer Water Company, Uyo. TORA Software was used in the analysis of the data using M – method. The result showed the values of the decision variables, 𝑥1,𝑥2,,𝑥3,𝑥4 and 𝑥5 to be 95, 0, 5.9, 10 and 17 respectively. Sensitivity Analysis of the problem was also discussed.

Keywords: Linear Programming, Optimum Production, Model, Maximization.

[1]. Ezema, B.I. and Amakon . U. (2012) Optimizing profit with the linear programming Model: A case on Boldem plastic industry limited enugu
[2]. Inter disciplinary Journal of Research in Business Vol 4 Is 2 pp 37-49
[3]. Fagoyinbo, I. S. and Ajibode, I. A (2010) Application of linear programming move Techinque in one effective use of resources for staff training
[4]. Frizzone J. et al (1997) linear programming model to optimize the water resource use in irrigation
[5]. Inter disciplinary Journal of Research in Business Vol 1 Pg 136-148
[6]. Khan, I. U., Bajuri, N.H. and Jadom I. A(2011) Optimal production levels for the different products manufactures at ICL a multinational Company in Pakistan
[7]. Nonso, A (2008) Application of linear programming for Managerial Decision (unpublished)
[8]. Sharma J. K. (1997) Operation Research; Theory and Application 4th Edition Macmillan Publishers india, Ltd. Pg.31-32.
[9]. Taha H. (1997) Operations Research; An introduction, Second Edition Macmillan Publishing Co. inc New York Pg 42-45.