iosr-Jm   Volume-18 ~ Issue-1 ~ Jan. – Feb. 2022

Abstract:An analytical study to determine the effect of certain material parameters on the rate of heat transfer and rate of mass transfer was made. The hydrodynamical equations of continuity, momentum, energy and concentration were non dimensionalized and solution obtained by perturbation method. Using numerical values, it was observed that increase in Prandtl number, radiation and heat absorption, depreciated the Nusselt number and skin friction of the nanofluid. Enhancement of the magnetic Hartmann number also enhanced the skin friction but increase in the frequency of oscillation depreciates skin friction and enhanced the rate of mass transfer. The Schmidt number depreciates both the skin friction and the Sherwood number as it is enhanced.......

Keywords: Rate of Heat transfer; Rate of mass transfer; Pertubation method, Nanofluids, Skin friction.

[1]. Seigel R, and Howell J R.( 1971) Thermal Radiation Heat Transfer. Student ed. Macgraw-Hill.
[2]. Lo, C.H, Tsung, T.T, Chen, L.C, Su, C.H and Lin, H.M (2005). Fabrication of Copper Oxide Nanofluid Using Submerged Arc Nanoparticle Synthesis System. Journal of Nanoparticle Research, 7: 313-320
[3]. Eastman, J.A, Choi, U.S, Li, S, Thompson, L.J and Lee, S (1997). Enhanced Thermal Conductivity Through the Development of Nanofluids. Materials Research Society. 3-11
[4]. Ngiangia, Alalibo.T and Orukari M.A. (2021); Heat Transfer Coefficient and Skin Friction Determination of Thermal Radiation Effects on MHD Convective Flow of Alumina Nanofluid Through a Non-Darcian Porous Plate. International Journal of Scientific and Engineering Research 12(5): 355-378
[5]. Ngiangia, Alalibo.T, Orukari M.A. Amadi, O and Nwabuzor, P. (2021); Onset of Transition to Non-Newtonian MHD Chemically Reacting Couette Copper Nanofluid Flow in a Radiative Porous Medium. International Journal of Mathematics Trends and Technology 67(5): 126 – 149 Doi: 10.14445/22315373/IJMTT-V6715P515

Abstract:Graph theory is one of the most important and basic topics of discrete mathematics in Mathematics. In all sectors of science graph theory has a great impact. The most common use of graphs occurs in Physics and Chemistry except Mathematics. It is also used in the modeling of Biology, Finance, and Computer science. Basic concepts of graphs are discussed here with classification with figures.
In this paper Historical background of graphs, classification, matrix representation of graphs, different types of graph operations, isomorphism, and some important theorems and problems are briefly reviewed. We will get a clear and visual concept on graph theory of discrete mathematics in this paper.

Key Word: Graphs, Matrix Representation, Incidence matrix, Adjacency Matrix, Cut-Set Matrix, Circuit Matrix, Graph operations.

[1]. Discrete Mathematics and its applications - Kenneth H. Rosen (7th edition)
[2]. Discrete Mathematics - Prof. Dr. Md. Ayub Ali and Prof. Dr. M.F Rahman
[3]. Graph theory - Harary
[4]. Graph Theory with Applications - J.A. Bondy and U.S.R. Murty
[5]. Graph Theory- Reinhard Diestel

Abstract: The mathematical modeling of cigarette smoking has an important role in understanding the transmission dynamics and developing effective prevention methods. In this study, a non-linear mathematical model is used to examine the extent of smoking from person to person and in a group of large social networks, as well as the social influence on temporary quit smokers. Firstly, the positivity and boundedness of the model solution are determined by the complete differential process, followed by the computation of the infection-free and endemic equilibrium of the system. The basic reproductive number is calculated by the next-generation matrix method. The local and global stability of the system is also presented.......

Keywords: Smoking model, social factor, smoking generation number, stability analysis, sensitivity analysis, numerical simulation.

[1]. World Health Organization Report on the Global Tobacco Epidemic, http ://whqlibdoc.who.int/publicatios/2009/9789241563918 eng full.pdf. 2009.
[2]. Australian Institute of Health and Welfare. 2004 National Drug Strategy Household Survey: detailed findings. Drug strategy series no.16, AIHW cat. no. PHE 66. Canberra: AIHW, 2005.
[3]. Michael Pearson, Lynn Michell Smoke Rings: social network analysis of friendship groups, smoking and drug-taking, Drugs: Education, Prevention and Policy, 2000, 7:1, 21-37.
[4]. Christakis NA, Fowler JH: The collective dynamics of smoking in a large social network. N Engl J Med 2008, 358:22492258
[5]. C.Castiho, Optimal control of an epidemic through educational campaigns, Electronic Journal of Differential Equations, 2006, no. 125: 1-11.

Abstract: Parallel computing is a type of computing architecture in which several processors execute or process an application or computation simultaneously. Parallel computing helps in performing large computations by dividing the workload between more than one processor, all of which work through the computation at the same time. Most supercomputers employ parallel computing principles to operate. Parallel computing is also known as parallel processing. This paper focuses on parallel programming in R, which is a programming language and software environment for statistical computing and graphics, there are several packages for parallel computing in R, some of which have existed a long time, e.g.Rmpi, nws, snow, sprint, foreach, multicore.R , which can also be compiled against multi-threaded linear algebra libraries, which helps to speed up calculations.

Key words: Parallel computing, computation simultaneously, programming language, statistical computing and graphics.

[1]. Brian Ripley [aut, cre, cph], Bill Venables [ctb], Douglas M. Bates [ctb], Kurt Hornik [trl] (partial port ca 1998), Albrecht Gebhardt [trl] (partial port ca 1998), David Firth [ctb], MASS: Support Functions and Datasets for Venables and Ripley's MASS,August 10 ,2021,< https://cran.r-project.org/web/packages/MASS/index.html>
[2]. Eddelbuettel, D, 2019, Parallel Computing With R: A Brief Review, August 15, 2021 <https://arxiv.org/abs/1912.11144>
[3]. Esam Mahdi , 2014, A Survey of R Software for Parallel Computing, American Journal of Applied Mathematics and Statistics, July 10,2021, <https://www.researchgate.net/publication/264518579_A_Survey_of_R_Software_for_Parallel_Computing>
[4]. Hadley Wickham [aut, cre], plyr: Tools for Splitting, Applying and Combining Data,August 15 ,2021, < https://cran.r-project.org/web/packages/plyr/index.html>
[5]. Hao Yu, Luke Tierney, Ulrich Mansmann, 2009, State-of-the-art in Parallel Computing with R, July10, 2021, <https://www.researchgate.net/publication/38105173_State_of_the_Art_in_Parallel_Computing_with_R>

Abstract: The Series of Sobolev inequality....

[1]. H. BREZIS , E H. LIEB , Sobolev Inequalities with Remainder Terms, Journal of Functional Analysis 62, 73-86 (1985)
[2]. A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl. 295(2004), 225-236..
[3]. G. Liu, Sharp k-order Sobolev inequalities in Euclidean space Rn and the sphere Sn, preprint 2006..
[4]. J. Dolbeault and G. Toscani. Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities. Int. Math. Res. Not. IMRN, (2):473–498, 2016.
[5]. M. Fathi, E. Indrei, and M. Ledoux. Quantitative logarithmic Sobolev inequalities and stability estimates. Discrete Contin. Dyn. Syst., 36(12):6835–6853, 2016

Abstract: In this paper, Lomax exponential distribution is considered for Bayesian analysis. The expressions for Bayes estimators of the parameter have been derived under squared error, precautionary, entropy, K-loss, and Al- Bayyati's loss functions by using quasi and gamma priors.

Keywords: Bayesian method, Lomax exponential distribution, quasi and gamma priors, squared error, precautionary, entropy, K-loss, and Al-Bayyati's loss functions.

[1]. Ijaz M, Asim SM, Alamgir, (2019): "Lomax exponential distribution with an application to real-life data". PLoS ONE 14(12): e0225827. https://doi.org/10.1371/journal.pone.0225827.
[2]. Zellner, A., (1986): "Bayesian estimation and prediction using asymmetric loss functions". Jour. Amer. Stat. Assoc., 91, 446-451.
[3]. Basu, A. P. and Ebrahimi, N., (1991): "Bayesian approach to life testing and reliability estimation using asymmetric loss function". Jour. Stat. Plann. Infer., 29, 21-31.
[4]. Norstrom, J. G., (1996): "The use of precautionary loss functions in Risk Analysis". IEEE Trans. Reliab., 45(3), 400-403.
[5]. Calabria, R., and Pulcini, G. (1994): "Point estimation under asymmetric loss functions for left truncated exponential samples". Comm. Statist. Theory & Methods, 25 (3), 585-600.