iosr-Jm   Volume-15 ~ Issue-5 ~ Sep-Oct 2019

Abstract: Fermat numbers are those in the form of Fn=22𝑛+1where n is a nonnegative integer. It was introduced by Pierre de Fermat, and the first four of these are primes. The search has been going on to see whether Fermat numbers are square-free, the mathematical term for a multiple of a square. This article tries to give a simple proof that Fermat numbers are square-free.

Key Word:Integer, square, square-free

[1]. Fermat‟s Number. https://en.wikipedia.org/wiki/Fermat_number
[2]. C. Clapham, J. Nicholson, The concise Oxford Dictionary of Mathematics. 4th ed. Oxford University Press, Oxford, 2009.
[3]. List of unsolved problems in mathematics. https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
[4]. Square-free integers. https://en.wikipedia.org/wiki/Square-free_integer
[5]. Peter J. Cameron. A Course on Number Theory. Page1-2. Chapter 1. Overview and revision. http://www.maths.qmul.ac.uk/~pjc/notes/nt.pdf

Abstract: One of the most elusive unsolved problems of today is Riemann hypothesis. For long mathematicians have struggled to prove this problem,and also tried to devise an elementary version of the problem, proof of which indirectly proves Riemann hypothesis. In 2002 J. C. Lagarias published such an elementary version of the hypothesis which has been widely accepted as an elementary equivalent of Riemann hypothesis. This article attempts to prove Lagarias's condition which consequently proves Riemann hypothesis....

Key Word: Harmonic number, natural logarithm, factorial.

[1]. Riemann hypothesis. https://en.wikipedia.org/wiki/Riemann_hypothesis
[2]. E. Bombieri(2000).The Riemann Hypothesis – official problem description(PDF), Clay Mathematics Institute.retrieved2008-10-25 Reprinted in (Borwein et al. 2008).
[3]. Riemann hypothesis. https://www.claymath.org/millennium-problems/riemann-hypothesis
[4]. K.O. Vlasov. Analysis of certain equivalents of Riemann hypothesis. DOI: 10.13140/RG.2.1.3078.0649. https://www.researchgate.net/publication/281399822_Analysis_of_certain_equivalent_for_the_Riemann_hypothesis
[5]. J. C. Lagarias (2002).An Elementary Problem Equivalent to the Riemann Hypothesis. The American Mathematical Monthly.109(6):534-543. Doi:10.1080/00029890.2002.11919883..

Abstract: Suppose that we are given a point, P, in the interior of a convex quadrilateral, Q, in the xy plane. Is there an ellipse, E₀, inscribed in Q and which also passes through P ? If yes, how many such ellipses ? By
inscribed in Q we mean that E₀ lies in Q and is tangent to each side of Q. Looked at in a dynamic sense: Imagine particle constrained to travel along the path of an ellipse inscribed in Q, so that the particle bounces off of each
side of Q along its path. Of course there are infinitely many such paths. Can we also specify a point in Q that the particle must pass through ? If yes, is such a path then unique ? We show below(Theorem 1) that the path is
unique when P lies on one of the diagonals of....

[1]. G. D. Chakerian, A Distorted View of Geometry, MAA, Mathematical Plums, Washington, DC, 1979, 130-150.
[2]. Jack Goldberg, "Matrix Theory with Applications", McGraw-Hill, 1991.
[3]. Alan Horwitz, Ellipses of maximal area and of minimal eccentricity inscribed in a convex quadrilateral, Australian Journal of
Mathematical Analysis and Applications, 2(2005), Issue 1, Article 4, 1-12.
[4]. Alan Horwitz, Dynamics of ellipses inscribed in triangles, Journal of Science, Technology and Environment, Volume 5, Issue 1
(2016), 1-21.
[5]. Morris Marden, The Location of the Zeros of the Derivative of a Polynomial, The American Mathematical Monthly, Vol. 42, No. 5
(May, 1935), pp. 277-286..

Abstract: Batch testing is a fundamental testing scheme that results into substantial saving in terms of cost and time. It's mainly applicable in cases with large population sizes and low prevalence rates. Studies on Batch testing have shown that Adaptive Batch testing is more efficient than Non-Adaptive Batch testing particularly as the number of stages increases. Most recent studies on Batch testing have shown that even with Truncation in inspection, Adaptive Batch testing remains more efficient. This study presents a Three-Stage Adaptive Batch testing Model with errors without Truncation with the view to establishing whether or not it's more efficient than the truncated estimator........

Key Word: Batch testing, Truncation, Sensitivity, Specificity, Efficiency

[1]. Dorfman R (1943). The detection of defective members of a large population, annals of mathematical statistics 14, 436-440.
[2]. Oliver-Hughes, JM and Swallow W.H (1994). A two stage adaptive group testing procedure for estimating small proportion. American statistical association. 89,982-993.[3]. Thompson K.M (1962). Estimating of proportion of vector in a natural population of insect's .Biometrics 18, 568-578.
[4]. Okoth.A.W.(2017). Multi-stage Pool Testing Model With Test Errors: Improved Efficiency .Group testing anew goal Estimation. biometrika, 52-181-194.
[5]. Wanyonyi W.R (2015). Estimation of proportion of a trait by batch testing with Errors in Inspection in a quality control process.

Abstract:In this present paper we have proved some Generalized of Integral Type C * -Valued Contraction with Fixed Point. Also we provided an example to support our main result..

Key Word:Metric Space, Algebra Valued Metric Space,C * -Algebra Valued Metric Space, Fixed Point Result, Contraction Mapping,Branciari Contraction Function.

[1]. S. Banach, Sur les ope´rations dans les ensembles abstraits et leur application aux equations inte'grales, Fund. Math. 3 (1922), 133-
181
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[3]. A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, International Journal of
Mathematics and Mathematical Sciences, 29 (2002),531-536.
[4]. KR. Davidson, C∗-Algebras by Example. Fields Institute Monographs, Vol. 6 (1996).
[5]. T. Kamran, M. Postolache, A. Ghiura, S. Batul, R. Ali, The Banach contraction principle in C.-algebra-valued b-metric spaces with
application. Fixed Point Theory Appl. 2016, (2016).

Abstract: The aim of this paper is to prove the generalized fuzzy metric space for four discontinuous mappings in non complete intuitionistic fuzzy metric spaces using contractive condition of integral type..

Key Word:Intuitionistic fuzzy metric spaces,weakly compatible mappig, common fixed point. AMS mathematics subject classification 47H10,54H25.

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[4]. GREGORI V., SAPENA A., On fixed point theorem in fuzzy metric spaces, Fuzzy Sets and Systems, 125(2002), 245-252.

Abstract: In this paper we consider nonlinear dynamical system to study the dynamics of malaria with relapse effect in both human and mosquito population. The total population is divided in to six compartments in which human population into three compartments and mosquito population into two compartments. We found the dynamical system has disease free equilibrium point and endemic equilibrium point. We also found that the basic reproduction number of the considered dynamical system.........

Key Word: Dynamics of Malaria,relapse effect,basic reproduction number, stability analysis, sensitivity analysis, numerical simulation.

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Abstract: In this work a study of the coexistence in the open ecological space with three species, where a prey is born, a natural predator and a prey-predator; here the normal model is treated, the equilibrium positions are analyzed and conclusions are drawn from future behavior. If a generalization of the model is made, the system is simplified reducing it to quasi-normal form making use of the analytical theory of differential equations; this facilitated the application of the second method of Liapunov to give conclusions of the future development regarding the coexistence.

Key Word: Ecology, Predator,Prey, Stability

[1]. Altair Santos de Oliveira Sobrinho, Camila Fogaça de Oliveira, Carolina MassaeKita, ÉricaRegina Takano Natti, Paulo Laerte Natti. "Modelagem Matemática e Estabilidade de Sistemas Predador-Presa"Universidade Estadual de Londrina, Londrina. (2016).
[2]. Antonio I Ruiz Chaveco, Sandy Sánchez Dominguez, Adolfo García. "Mathematical Modeling of Polemerization of Hemoglobin S". Ed. Lab Lambert. 2015.
[3]. Colin R. Townsend, Michel Begon e John L. Harper "Fundamentos em Ecologia" 3ra Edição. Porto Alegre, (2010).
[4]. Dajoz, Roger "Principios de Ecologia" 7ma, Edição, Porto Alegre. (2005).
[5]. López, J. M., &Blé, G. G. Modelo Depredador - Presa. (2008). Revista de Ciencias Básicas UJAT, 7 (2), 25 - 34.

Abstract: In this article weestablishthe application for viscosity approximation methods for Nonexpansive sequence of mappings. we associate converges strongly to a common element of the set of fixed points of sequence of mappings and also the set of solutions of the Variational inequality for an inverse strongly-monotone sequence of mappings which solves some Variational inequality

Key Word: Viscosity approximation; fixed point; Inverse-strongly monotone mapping; Nonexpansive mapping; Variational inequalities

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[4]. A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46–55.
[5]. H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291.

Abstract: In this paper, a mathematical modelling of Onchocerciasis also known as Riverblindness epidemics is discussed. Approximate analytical solutions for the non-linear equations in Onchocerciasis epidemics are obtained by using the Homotopy analysis method (HAM). For this purpose, a theorem is proved to show the convergence of the series solution obtained from the proposed method. Analytical expressions pertaining to the number of susceptible and infected individuals are derived for all possible values of parameters. Furthermore, we present a condition enabling the homotopy analysis method (HAM) to converge to the exact solution of the nonlinear differential equations.

Key Word: Onchocerciasis epidemics, Mathematical modeling, Homotopy analysis method (HAM), Convergence, Nonlinear differential equations

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[5]. D. J. Jeffrey, S. Liang, Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation, Commun. Nonlinear Sci. Numer. Simulat. 14, 40574064 (2009)..