iosr-Jm   Volume-15 ~ Issue-2 ~ March-April 2019

Abstract: In this paper, we analyze the existing rules for constructing derivatives of the scalar and tensor functions of the tensor argument with respect to the tensor argument and the theoretical positions underlying the construction of these rules. We perform a comparative analysis of these rules and the results obtained in the framework of these rules. Considering the existing approaches, we pay due attention to the earliest of them which for some reason is not reflected in later publications on the issue under consideration, and we give to this approach the further development. The rules for constructing the derivatives..........

Key Word: Differentiation with respect to a tensor, Rules for differentiation and forms of derivatives, Scalar and tensor functions of tensor argument

[1]. A.A. Rogovoy, Formalized approach to construction of the state equations for complex media under finite deformations, Continuum
Mechanics and Thermodynamics, 24, 2012, 81-114. http://dx.doi.org/10.1007/s00161-011-0220-y.
[2]. A. Rogovoy and O. Stolbova, Modeling the magnetic field control of phase transition in ferromagnetic shape memory alloys,
International Journal of Plasticity, 85, 2016, 130-155. http://dx.doi.org/ 10.1016/j.ijplas.2016.07.006.
[3]. B.E. Pobedrya, Lectures on Tensor Analysis (Moscow: Moscow State University, 1974, in Russian).
[4]. P.K. Rashevsky, Riemannian Geometry and Tensor Analysis (Moscow: Nauka, 1967, in Russian).
[5]. I.S. Sokolnikov, Tensor Analysis (Moscow: Nauka, 1971, in Russian).

Abstract: By having used of differential subordination, it has been investigated in the present paper, subordination relations, inclusion relations, distortion theorem and inequality properties are discussed of the class𝕄𝔹(𝜶,𝝀,𝓵,𝝁,𝜜,𝜝). In this paper it has been introduced some new classes 𝕄𝔹(𝜶,𝝀,𝓵,𝝁,𝜜,𝜝) of meromorphic functions which are defined by means a meromorphic function using a new operator.

[1]. Aouf, M. K. and Mostafa, A. O. (2008) Certain subclass of p-valent meromorphic functions involving certain operator, J. Inequal. Pure and Appl. Math.,, (9), Article 45, 8pp.
[2]. Aqlan, E., Jahangiri, J. M. and Kulkarni, S. R. (2003) Certain integral operator applied to meromorphic p-valent functions, J. of Nat. Geom., 24, 111-120.
[3]. Dziok, J. and Srivastava, H. M. (1993) Classes of analytic functions associated with the generalized hypergeometric functions, Appl. Math. Comput., 103, 1-13.
[4]. Dziok, J. and Srivastava, H. M. (2002) Some subclasses of analytic functions with fixed arguments of coefficients associated with generalized hypergeometric function. Adv. Stud, Contemp. Math., 5, 115-125.
[5]. Dziok, J. and Srivastava, H. M. (2003) Certain subclasses of analytic functions associated with the generalized hypergeometric functions. IntegralTrans. Spec. Funct., 14, 7-18.

Abstract: The aim of this article was to compare Pearson's Chi-square to Uniform (U), Column (C), Row (R), and R+C ordinal contingency tables models. Data on gender, university attended for B.Sc., B.Sc. and M.Sc. grades of 116 M.Sc. graduates were collected from Department of Statistics, University of Ilorin, Nigeria. Model estimation was carried out by maximum likelihood method and goodness of fit was assessed by likelihood ratio statistic. Pearson's chi-square rejected the null hypothesis of independence in all cases; the U model rejected in 2 of 6 cases while R rejected in 4 cases. The C model rejected in 3 cases while R+C rejected in 5 out of 6 cases. Pearson's chi-square reached same..........

Keywords: Contingency table, Chi-square, Row model, Colum model, Likelihood ratio

[1]. Agresti, A. (2007). An Introduction to Categorical Data Analysis (2nd ed.). New Jersey: John Wiley.
[2]. Aktas, S. & Saracbasi, T. (2003). Analysis of triangular contingency tables. Hacettepe J. of Math. & Stat., 32, 43-51.
[3]. Altunay, S.A. & Saracbasi, T. (2009). Estimation of symmetric disagreement using a uniform association model for ordinal
agreement data. ASTA Advances in Sta. Anal., 93(3), 335-343.
[4]. Camminatiello, I., D'Ambra, A. & Sannacchiaro, A. (2014). The association in a two-way contingency table through log
odds ratio analysis: the case of Sarno river pollution. Springer Plus, 3. DOI 10.1186/2193-1801-3-384.
[5]. Eshima, N., Tubata, M. & Tsujitani,, M. (2001). Property of the RC(M) association model and a summary measure of
association in the contingency table. J. Japan Stat. Soc., 31(1), 15-26..

Abstract: In this paper, the controllability of a class of nonlinear integro-differential equation with implicit derivatives is investigated. We employ the Darbo fixed point theorem to investigate our result. Also , we give example to illustrate to obtained result.

Keyword: Nonlinear integrodifferential equation, Darbo fixed point theorem, controllability

[1]. Aghajani, A., Banas, J., andSabzali, N. (2013). Some generalizations of Darbo fixed point theorem and applications. Bulletin of the
Belgian Mathematical Society-Simon Stevin, 20(2); 345-358.
[2]. Balachandran, K., and Dauer, J. P. (2002). Controllability of nonlinear systems in Banach spaces: a survey. Journal of Optimization
Theory and Applications, 115(1): 7-28.
[3]. Balachandran, K., andSomasundaram, D. (1983). Controllability of a class of nonlinear systems with distributed delays in
control. Kybernetika, 19(6): 475-482.
[4]. Balachandran, K., and Somasundaram, D. (1985). Relative controllability of nonlinear systems with time varying delays in
control. Kybernetika, 21(1): 65-72.
[5]. Balachandran, K., andSomasundaram, D. (1986). Controllability of nonlinear delay systems with delay depending on state
variable. Kybernetika, 22(5): 439-444..

Abstract: In this paper, we consider a SEIR epidemic model with homogenous transmission function and treatment. Found the basic reproduction number 0 R and equilibrium points namely disease-free equilibrium and endemic equilibrium. The global stability of the disease free equilibrium and endemic equilibrium is proved using Lyapunov function and Poincare-Bendixson theorem plus Dulac's criterion respectively and also study the sociological and psychological effect on the infected population. We gave some numerical result to analyze our model with actual model..........

Keywords: Routh Hurwitz criteria, Lyapunov function, Dulac's criterion, Basic reproduction number, Stability,
Treatment.

[1]. H. Laarabi, E. Labriji, M. Rachik, A. Kaddar.(2012). optimal control of an epidemic model with a saturated incidence rate,
Nonlinear Analysis Modelling and Control, 17(4),448-459.
[2]. H.W. Hethcote, P.van driessche.(1991).some epidemiological models with nonlinear incidence, J.Math. Biol, 29, 271-287.
[3]. O. Adebimpe, L.M. Erinle-Ibrahim, A.F.Adebisi.(2016). stability analysis of SIQS epidemic model with saturated incidence rate.
Applied Mathematics,7,1082-1086.
[4]. S.G. Ruan, W.D.Wang.(2003). dynamical behavior of an epidemic model with a nonlinear incidence rate,
J.Differ.Equations,188,135-163.
[5]. S. Pathak, A. Maiti, and G.P. samanta.(2010). rich dynamics of an SIR epidemic model. Nonlinear Analysis Modelling and
Control, 15,71-81..

Abstract: The effects of magnetic field on boundary layer flow with heat and mass transfer of a water-based nanofluid containing gyrotactic microorganisms over a vertical plate are numerically investigated.The governing boundary layer equations are formulated and transformed into ordinary differential equations using a suitable similarity transformation.The resulting ordinary differential equations are solved numerically using the fourth order Runge-kutta method with shooting technique. Pertinent results are presented graphically and discussed quantitatively with respect to variation in the controlling parameters such as; bioconvection lewis number (Lb),traditional lewis number (Le),bioconvection peclet lewis number...............

[1]. Alfen,H.L(1942),Existence of electromagnetic Hydrodynamic waves, Nature vol.150,pp 405−406
[2]. Aziz A.(2010) ,Hydrodynamic and thermal slip flow boundary layers over a flat surface with constant heat flux boundary condition, Communications in nonlinear simulations,15:573-580
[3]. Aziz A. Khan and W.A. Pop I. (2012)Free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms ,Int. J. Therm. Sci. vol.56, pp 48− 57.
[4]. Aziz A. and Khan. W.A (2012) Natural convective boundary layer flow of a nanofluid past a convectively heated vertical plate, International Journal of Thermal Sciences vol.52 pp 83-90.
[5]. Bachok N. and Ishak A., Pop I. (2010) Boundary layer flow of nanofluids over a moving surface in a flowing fluid. International Journal of Thermal Sciences, 49, 1663-1668.

Abstract: Practical optimization problems frequently involve nonlinear behaviors that must be taken into account. Approaches to nonlinear optimization problems often utilize approximation of complicated functions by simpler ones which are easier to calculate, and which show the relations between the variables more clearly. In this paper, we obtained an approximate optimal solution to a convex quadratic objective function by quadratic approximation. This approach was applied to a real-world numerical example to obtain the optimum which is the same as that from the conventional method.

Keywords: Objective function approximation, Optimal solution, Quadratic approximation, Nonlinear optimization programs.

[1]. L. Jones, "A simple Lemma on greedy approximation in Hilbert space and convergence rates for projection pursuit
regression and network training," The annals of statistics, Vol. 20, pp. 608-613, 1992.
[2]. V. N. Temlyakov, "On estimates of Approximation numbers and best Bilinear Approximation," Constructive Approximation, Vol.
8, pp. 23-33, 1992.
[3]. P. M. Gruber, "Asymptotic estimates for best and stepwise approximation of convex bodies II," Forum Mathematics, Vol. 5, pp.
521-538, 1993.
[4]. Jr. K. Boroczky and M. Ludwig, "Approximation of convex bodies and a momentum Lemma for power diagrams,"
Monatsh Mathematics, Vol. 127, pp. 101-110, 1999.
[5]. H. Pottmann, R. Krasauskas, B. Hamann, K. Joy, and W. Seibold, "On piecewise Linear Approximation of Quadratic
functions," Journal of Geometry and Graphics, Vol. 4 (1), pp. 31-53, 2000.

Abstract: Thermal protective clothing not only has the performance of ordinary protective clothing, but also has the function of protecting the human body under high temperature conditions[3]. It uses mathematical methods to study thermal protective clothing, aiming to reveal the heat transfer law inside the thermal protective clothing. Provide a theoretical basis for the development of thermal protective clothing[1][2][4][5][6]. At a certain ambient temperature, a partial differential equation model was established by using the heat conduction model to study the heat transfer model of the external temperature through heat protection clothing to the dummy skin under high temperature environment and solved by finite difference method.

Keywords: heat conduction, partial differential equation, finite difference method

[1]. Pan Bin. Thermal protection clothing heat transfer mathematical modeling and parameter determination inverse problem [D].
Zhejiang Institute of Technology. 2017.
[2]. Lu Linzhen. Heat transfer model and optimal parameters of multi-layer thermal protective clothing [D].Zhejiang Institute of
Technology. 2018.
[3]. Zhu Fanglong. Thermal protection of clothing. Beijing: China Textile Press, 2015.10.
[4]. Liu Yaya. The heat source inversion problem of one-dimensional heat conduction equation [D].Southeast University, 2016.
[5]. Liu Ziwen, PuChunde, Xie Liang, Li Bin, Yang Dashuai, Pan Dongyu. Correlation between temperature and heating time under
one-dimensional heat conduction conditions[J]. Frontiers of Earth Science, 2016, 6(2): 72-78.