iosr-Jm   Volume-15 ~ Issue-4 ~ July-August 2019

Abstract: The properties of the nilpotent Cayley graph 𝐺 𝑍𝑛,𝑁 associated with the set of nilpotent elements of the residue class ring 𝑍𝑛,⨁,⊙ is studied by the authors. The vertex cover, the vertex dominating set and the related domination parameters of these graphs are determined in this paper.

Key Word: Nilpotent element, Symmetric set, Cayley graph, Nilpotent Cayley graph

[1]. Allan, R. B, Laskar, R.: On domination and independent domination number of a Graph, Discrete Math, 23: 73-76 (1978).
[2]. Allan, R. B., Lasker, R., Hedetniemi, S.T.: A note on total domination, Discrete Math, 49: 7-13(1984).
[3]. Apostol, T. M.: Introduction to Analytic Number Theory, Springer International Student Edition (1989).
[4]. Berge, C.: Theory of Graphs and its Applications, Methuen, London (1962).
[5]. Bondy, J.A., Murty, U.S.R.:Graph Theory and related topics, Macmillan, London(1979)..

Abstract: Earlier many researchers devoted to describe the nature of traffic flow using PDE and ODE mostly using the concept of conservation law in fluid mechanics. Here we have described the nature of vehicles flow using nonlinear dynamical system. We have formulated a new model that show us the causes of delay time as a consequence of many vehicles involved in the way. That is, the qualitative behavior of the flow of vehicles with inflow, outflow, and blocking effects was presented. We observed freely flow vehicles move with the allowable speed whereas blocked vehicles move with the restricted speed to reduce congestions on the road. The rate of flow is high for free vehicles and..........

Key Word: Nonlinear dynamical systems, Equilibrium point, Traffic flow, Stability analysis, Retardation number, Well-Posedness,Sensitivity analysis.

[1] M. Caramia, C. D'Apice, B. Piccoli, and A. Sgalambro.Fluidsim: A Car Traffic Simulation Prototype Based on Fluid Dynamic, Algorithms, 3 (2010), 294–310.
[2] R.E. Chandler, R. Herman and E. W. Montroll. Traffic Dynamics: Studies in Car Following, Operations Research, 6 (1958), 165–184.
[3] G. F. Newell. Nonlinear Effects in the Dynamics of Car Following, Operations Research, 9 (1961), 209–229.
[4] M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama. Dynamical Model of Traffic Congestion and Numerical Simulation, Phys. Rev. E, 51 (1995), 1035–1042.
[5] H M. Zhang. A Non-equilibrium Traffic Model Devoid of Gas-like Behavior, Transportation Research Part B: Methodological, 36 (2002), 275–290..

Abstract: Malaria is an infectious disease caused by the Plasmodium parasite and is transmitted between humans through bites of female anopheles' mosquitoes. The disease continues to emerge in developing countries and remains as a global health challenge. In this paper, a mathematical model is formulated that insights in to some essential dynamics of malaria transmission with environmental management strategy for malaria vector control, insecticide treated bed net, indoor residual spray and treatment with antimalaria drugs as control strategies for humans so as to minimize the disease transmission or spread. The reproduction numbers with single and combined......

Key Word: Endemic malaria, Infectious diseases, Numerical simulation, Plasmodium parasite, Reproduction number

[1]. Centers for Disease Control and Prevention. CDC - Malaria (accessed August 8, 2011) http: //www.cdc.gov/MALARIA.
[2]. WHO, "Investing in health research for development", Technical Report, World Health Organization, Geneva, 1996.
[3]. Tumwiine J., Mugisha J., Luboobi L., "A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity", Journal of Applied Mathematics and Computation, vol. 189, pp. 1953-1965, 2005.
[4]. Global Strategic framework for Integrated Vector Management. Geneva, WHO, 2004.
[5]. DOMC, National Guidelines for Diagnosis, Treatment and Prevention of Malaria in Kenya, Division of Malaria Control, Ministry of public health and sanitation, Nairobi, Kenya,3rd edition, 2010.

Abstract: Quasi-Newton Method is used to solve no-linear optimization problems as an extension of Newton Method for multi-variables. This method is also used to solve second order derivatives of the o f the objective functions in the form of Hessian Matrix. Quasi-Newton Method is the most effective method for finding minimizers of a smooth non-linear function when second derivatives are either unavailable or too difficult to compute. The algorithm method is used on the functions to generate solutions and theses solutions are found to be convergent after certain number of iterations. The new point is then obtained by the sum of the previous point and the result is multiplied by the step length and the search direction. This process is continued until convergent is reached and this method is easy to handle . The Qusi-Newton Method uses less time for computations and less number of iterations.

Key Word:Quasi, optimization, minimization, Matrix, symmetric, Genetic Algorithm, Particle Swarm and positive definite.

[1]. Deufhard. P. (2010): The History of Newton's Methods. Informatics Techlink, 55-83.
[2]. Fetcher.R. (1987): Practical Methods of Optimization . International Journal of Optimization Theory, 100-170.
[3]. Forsgren. A, Gill. P.E and Wright. M.H (2003): Interior Methods for Non Linear Optimization, SIAM 44(4) , 525-597.
[4]. Hanke. M. (1998): Conjugate Gradient type Method for ill-posed Problems, Pitman Research Notes in Mathematics Series, Vol
327.
[5]. Heyvan . A and Ashraf. G. (2010): A New Structured Quasi-Newton Algorithm using Partial Differentiation on Hessian Matrix.
Journal of Computational and Applied Mathematics, 805=811..