iosr-Jm   Volume-11 ~ Issue-1 ~ Jan-Feb 2015

Abstract: In this paper we introduce the notion of δ ˆ –closed sets and studied some of its basic properties and characterizations. It shows this class lies between –closed sets and g–closed sets in particularly lies between - I–closed sets and g–closed sets. This new class of sets is independent of closed sets, semi closed and –closed sets. Also we discuss the relationship with some of the known closed sets.

Keywords and Phrases: δ ˆ –closed, δ ˆ –open.

[1]. Abd El-Monsef, M.E., S. Rose Mary and M.Lellis Thivagar, On G ˆ
-closed sets in topological spaces, Assiut University Journal of
Mathematics and Computer Science, Vol 36(1), P-P.43-51(2007).
[2]. Akdag, M. -I-open sets, Kochi Journal of Mathematics, Vol.3, PP.217-229, 2008.
[3]. 2a) Andrijevic, D. Semi-preopen sets, Mat. Vesnik, 38 (1986), 24-32.
[4]. Arya, S.P. and T. Nour, Characterizations of S-normal spaces, Indian J. Pure. Appl. Math. 21(8) (1990), 717-719.
[5]. Bhattacharya, P. and B.K. Lahiri, Semi-generalized closed sets in topology, Indian J. Math., 29(1987), 375-382.
[6]. Dontchev, J. and M. Ganster, On -generalized closed sets and T3/4-spaces, Mem. Fac. Sci. Kochi Univ. Ser. A, Math., 17(1996),
15-31.
[7]. Dontchev, J., M. Ganster and T. Noiri, Unified approach of generalized closed sets via topological ideals, Math. Japonica, 49(1999),
395-401.

Abstract: In this paper we analyze the comparison of queuing models to vehicular traffic at kanyakumari district in different places.This section introduces the data sources discuss the M/M/1 and M/D/1 queuing models which this article uses to model vehicular traffic could be minimized using queuing theory in kannyakumari district .The result showed that traffic intensity 𝜌<1.This paper compares the result obtained from both methods and describes how these data collected at various places in Kanyakumari district.

Keywords: M/M/1 queuing model, M/D/1 queuing model, probability distribution,Queuing theory, Poisson process.

[1]. Wayne L. Winston, Introduction to Probability Models, USA, Thomson Learning, 2004, Chap 8, pp 308-388
[2]. X., Zhou, X. and List, G.F. (2011). Using time-varying Tolls to optimize truck arrivals at ports, Transportation Research, Part E: Logistics and Transportation Review47(6): 965–982.
[3]. O.J. Boxma and I.A. Kurkova. The M/G/1 queue with two service speeds. Advances in Applied Probability, 33:520–540, 2001.
[4]. J.Y. Cheah and J.M. Smith. Generalized M/G/C/C state dependent queuing models and pedestrian traffic flows. Queuing Systems, 15:365–385, 1994.
[5]. a c b c sundarapandian, V .(2009). "7th queuing theory ", Probability, statics and queuing theory. PHI Learning ISBN 8120338448.

Abstract: Let 𝑋 be a real or complex Banach space and 𝑌 be a normed linear space. Suppose that 𝑓:𝑋→𝑌 is a Frechet differentiable function and 𝐹:𝑋⇉2𝑌 is a set-valued mapping with closed graph. Uniform convergence of Chord method for solving generalized equation 𝑦∈𝑓 𝑥 +𝐹 𝑥 …….(∗), where 𝑦∈𝑌 a parameter, is studied in the present paper. More clearly, we obtain the uniform convergence of the sequence generated by Chord method in the sense that it is stable under small variation of perturbation parameter 𝑦 provided that the set-valued mapping 𝐹 is pseudo-Lipschitz at a given point (possibly at a given solution). Keywords: Chord method, Generalized equation, Local convergence, pseudo-Lipschitz mapping, Set-valued mapping. AMS (MOS) Subject Classifications: 49J53; 47H04; 65K10.

[1]. S.M. Robinson, Generalized equations and their solutions, Part I, Basic theory, Math. Program. Stud., 10, 1979, 128–141.
[2]. S.M. Robinson, Generalized equations and their solutions, part II: Application to nonlinear programming, Math. Program. Stud., 19, 1982, 200–221.
[3]. M.C. Ferris and J.S. Pang, Engineering and economic applications of complementarily problems, SIAM Rev., 39, 1997, 669–713.
[4]. A. L. Dontchev, Uniform convergence of the Newton method for Aubin continuous maps, Serdica Math. J., 22 , 1996, 385–398.
[5]. A. Pietrus, Does Newton's method for set-valued maps converge uniformly in mild differentiability context?, Rev. Columbiana Mat., 34, 2000, 49–56.
[6]. M.H. Geoffroy and A. Pietrus, A general iterative procedure for solving nonsmooth generalized equations, Comput. Optim. Appl., 31(1), 2005, 57–67.

Abstract: In this paper, Bayesian estimation using diffuse (vague) priors is carried out for the parameters of a two parameter Weibull distribution. Expressions for the marginal posterior densities in this case are not available in closed form. Approximate Bayesian methods based on Lindley (1980) formula and Tierney and Kadane (1986) Laplace approach are used to obtain expressions for posterior densities. A comparison based on posterior and asymptotic variances is done using simulated data. The results obtained indicate that, the posterior variances for scale parameter  obtained by Laplace method are smaller than both the Lindley approximation and asymptotic variances of their MLE counterparts.

Keywords: Weibull distribution, Lindley approximation, Laplace approximation, Maximum Likelihood Estimates

[1]. Cohen, A.C. (1965): Maximum Likelihood Estimation in the Weibull Distribution Based on Complete and Censored Samples.
Technometrics, 7, 579-588.
[2]. Lindley, D.V. (1980). Approximate Bayesian Method, Trabajos Estadistica, 31, 223-237.
[3]. Tierney L, Kass, R.E. and Kadane, J.B. (1989): Fully exponential Laplace approximations to expectations and variances of nonpositive
functions. Journal of American Statistical Association, 84, 710-716.
[4]. Tierney L. and Kadane, J.B. (1986): Accurate Approximations for Posterior Moments and Marginal Densities. Journal of American
Statistical Association, 81, 82-86.

Abstract: In this paper we provide the asymptotic expansion of the roots of nonlinear dynamic system with two delays. We develop a series expansion to solve for the roots of the nonlinear characteristic equation obtained from the HIV-1 dynamical system. Numerical calculation are carried out to explain the mathematical conclusions.
Keywords: Asymptotic expansion, Delay Differential equations, Eigenvalues, HIV-1.

[1] S. Bonhoeffer, R.M. May, G.M Shaw, M.A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94, 1997, 6971 - 6976. [2] X.Y. Song, A.U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329, 2007, 281 - 297. [3] M.A. Nowak, S. Bonhoeffer, G.M. Shaw, R.M. May, Anti-viral drug treatment: dynamics of resistance in free virus and infected cell populations, J. Theor. Biol., 184, 1997, 203 - 217. [4] T.B. Kepler, A.S. Perelson, Drug concentration heterogeneity facilitates the evolution of drug resistance, Proc. Natl. Acad. Sci. USA, 95, 1998, 11514 - 11519. [5] P.K. Roya, A.N. Chatterjee, D. Greenhalgh, J.A. Khanc, Long term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model, Nonlinear Analysis: Real World Applications, 14, 2013, 1621 - 1633. [6] R.J. Smith, L.M. Wahl, Distinct effects of protease and reverse transcriptase inhibition in an immunological model of HIV-1 infection with impulsive drug effects, Bull. Math. Biol., 66, 2004, 1259 - 1283. [7] M. Pitchaimani, C. Monica, M. Divya, Stability analysis for HIV infection delay model with protease inhibitor, Biosystems, 114, 2013, 118 - 124.

Abstract: This paper presents the study of the effects of variable thermal conductivity and Darcy-Forchheimer on magnetohydrodynamic free convective flow in a vertical channel in the presence of constant suction. The resulting governing equations are non-dimensionalised, simplified and solved using Crank Nicolson type of finite difference method. To check the accuracy of the numerical solution, steady-state solutions for velocity, temperature and concentration profiles are obtained by using perturbation method. Numerical results for the velocity, temperature and concentration profiles are illustrated graphically while the skin friction, Nusselt number and Sherwood number are tabulated and discussed for some selected controlling thermo physical parameters involved in the problem to show the behavior of the flow transport phenomena. It is found that the velocity and temperature increased due to increase in variable thermal conductivity parameter and there is decrease in concentration due to increase in Darcy-Forchheimer number. It is also observed that the numerical and analytical solutions are found to be in excellent agreement.

1]. M. Abdou, Effect of radiation with temperature dependent viscosity and thermal conductivity on unsteady stretching sheet through
porous medium, Non-linear Analysis: Modeling and Control, 15(3), (2010), 257-270.
[2]. B. Carnahan, H. A. Luther and J. O. Wilkes, Applied numerical methods, John Wiley and Sons, New York, (1996).
[3]. T. C. Chiam. Heat transfer in a fluid with variable thermal conductivity over a linearly stretching sheet, Acta Mechanic, 129,
(1998), 63-72.
[4]. P. Gitima, Effects of variable viscosity and thermal conductivity of micropolar fluid in a porous channel in presence of magnetic
field, International Journal for Basic Sciences and Social Sciences, 1(3), (2012), 69-77.
[5]. M. A. Hossain, M. S. Munir and I. Pop, Natural convection with variable viscosity and thermal conductivity from a vertical wavy
cone, International Journal of Thermal Sciences, 40, (2001), 437- 443.
[6]. D. B. Ingham and I. Pop, Transport phenomena in porous media, Elsevier, Oxford, UK. (2005).
[7]. R. Kandasamy, K. Perisamy and P. K. Sivagnana, Effects of chemical reaction, heat and mass transfer along a wedge with heat
source and concentration in the presence of suction or injection, International Journal of Heat and Mass Transfer, 48, (2005,) 1388-
1394.

Abstract: We begin with some preliminaries on measure-theoretic probability theory which in turn, allows us to discuss the definition and basic properties of martingales. We then discuss the idea of stopping times and stopping process. We state auxiliary results and prove a theorem on a stopping process using the Càdlàg property of the process.
Keywords: Càdlàg, martingale, stopping process, stopping time

[1]. Doyle, P. G. and Snell, J. L. (1984) Random Walks and Electrical Networks. Carus Mathematical Monographs, Mathematical
Association of America Washington DC
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[6]. Gusak, D., Kukush, A. and Mishura Y, F. (2010). Theory of Stochastic Processes. Springer verlag.
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Abstract: The problem we address is to decide optimal ordering policy in a two echelon inventory system with repairable items. The system consists of a set of operating sites which are base station, depot. Each operating site has an inventory spare items to run the system properly without any delay of supplying items against failure demand. Replenishment of stock at base station is done from depot. The base station contains an inspection center which is used to inspect the failed items. The arrival of a failed items follows Poisson process with parameter  (>0) and the inspection time follows an exponential distribution with rate  (>0). After inspection
the repairable items are sent to depot for repair, otherwise the item is treated as condemned and it would be removed from the system. The repaired items which can be used are included in the depot stock. The system is designed under the continuous review Markov process model to find the steady distribution and applying the decision rule to obtain the optimal average cost of the system and optimal ordering policy by using Markov decision process concept. Numerical examples are provided to illustrate the problem.
Keywords: Repairable items, Poison arrival, Inspection time, Transition probability, Value iteration.

[1]. Allen, S.G. and D'Esopo, D.A., An Ordering Policy for Repairable Stock Items. Operations Research, 16. 1968, 669-674.
[2]. Axs a ter, S., Simple Solution Procedures for a Class of Two-Echelon Inventory Problems. Operations Research, 38. 1990, 64-69.
[3]. Donald Gross and Harris, M., Fundamentals of Queueing Theory, 4th Edition, John Wiley and Sons Ltd, England. 2008.
[4]. Hau L. Lee. and Kamran Moinzadeh., A Repairable Item Inventory System with Diagnostic and Repair Service. European of
Operational Research , 40(2), 1989, 210- 221.
[5]. Hopp, W.J., Zhang, R.Q. and Spearman, M.L., An Easily Implementable Hierarchical Heuristic for a Two-echelon spare
Distribution System. IIE Transactions, 31, 1999, 977-988.
[6]. Howard, R.A., Dynamic Programming and Markov Processes. John Wiley and Sons, Inc:, New York. 1960.
[7]. Jun Xie Hongwei Wang., Optimization Framework of Multi-Echelon Inventory System for Spare Parts. Wireless Communications,
Networking and Mobile computing, WiCOM'08.
th 4 International Conference, 2008, 1-5.

Abstract: In this paper, a numerical solution based on Adomian Decomposition Method (ADM) is used for finding the solution of higher order nonlinear problem which arise from the problems of calculus of variations. This approximation reduces the problem to an explicit system of algebraic equations. One numerical example is also given to illustrate the accuracy and applicability of the presented method.

Keywords: Adomian decomposition method, Calculus of variations, Haar wavelet series, Higher order nonlinear problem, Single-term Haar wavelet series method.

[1]. M. Dehghan and M. Tatari, The use of Adomian decomposition method for solving problems in calculus of variations, Math. Probl. Eng. 2006, 2005 1-12.
[2]. J. Gregory and R. S. Wang, Discrete variable methods for the dependent variable nonlinear extremal problems in the calculus of variations, SIAM Jour. of Numeric. Ana. 2, 1990, 470–487.
[3]. Mohammad Maleki and Mahmoud Mashali-Firouzi, A numerical solution of problems in calculus of variation using direct method and nonclassical parameterization, Journal of Computational and Applied Mathematics, 234, 2010, 1364-1373
[4]. A. Saadatmandi and M. Dehghan, The numerical solution of problems in calculus of variation using Chebyshev finite difference method, Phys. Lett. A, 372, 2008, 4037-4040.
[5]. S. Sekar and K. Prabakaran, Numerical solution of nonlinear problems in the calculus of variations using single-term Haar wavelet series, International Journal of Mathematics and Computing Applications, 2(1), 2010, 25-33.
[6]. S. Sekar and M. Nalini, Numerical Analysis of Different Second Order Systems Using Adomian Decomposition Method, Applied Mathematical Sciences, 8(77), 2014, 3825-3832.
[7]. S. Sekar and A. Kavitha, Numerical Investigation of the Time Invariant Optimal Control of Singular Systems Using Adomian Decomposition Method, Applied Mathematical Sciences, 8(121), 2014, 6011-6018.
[8]. M. Tatari, M. Dehghan, Solution of problems in calculus of variations via He's variational iteration method, Phys. Lett. A, 362, 2007, 401-406.