i-manager's Journal on Mathematics (JMAT)


Volume 7 Issue 2 April - June 2018

Research Article

Proof of Beal's Conjecture and Fermat Last Theorem using Contra Positive Method

Vinay Kumar*
Professor, Vivekananda School of IT, VIPS, GGSIPU, New Delhi, India.
Kumar. V. (2018). Proof of Beal's Conjecture and Fermat Last Theorem using Contra Positive Method. i-manager’s Journal on Mathematics, 7(2), 1-7. https://doi.org/10.26634/jmat.7.2.14127

Abstract

“If Ax + By = Cz , for integers A, B, C ≥2 and integers x, y, z greater than 2 , then A, B, C must have a common prime factor”. The statement is known as Beal's conjecture (Rubin & Silverberg, 1994). Without loss of generality, integers B and C can be expressed in terms of A. Assuming B = A + m and C = A + n, the present study proves the conjecture for all the four cases: i) m = 0, n = 0; ii) m = 0, n≠ 0; iii) m≠0, n = 0; and iv) m≠0, n ≠0. A, B, and C can be ordered (sequenced) in six different ways. A theorem that is proved for one sequence, the same theorem can easily be proved for other five sequences. Contrapositive approach together with integer division algorithm is used to prove the conjecture. Contrapositive statement of Beal's x conjecture is “if A, B, and C have no common prime factor then no integers A, B, C and integers x, y, z > 2 such that Ax + By = Cz”. Some basic and fundamental properties of quadratic equation are also used in the proof.

Research Paper

Strong and Δ - Convergence Theorems using Modified Proximal Point Algorithm in CAt(0) Spaces

Apurva Kumar Das* , Shailesh Dhar Diwan**, Samir Dashputre***
* Lecturer, Department of Mathematics, Government Polytechnic Sukma, Chhattisgarh, India.
** Associate Professor, Department of Mathematics, Government Engineering College, Raipur, Chhattisgarh, India.
*** Assistant Professor, Department of Mathematics, Government College Arjunda, Balod, Chhattisgarh, India.
Das. A.K., Diwan. S.D., and Dashputre. S. (2018). Strong and Δ - Convergence Theorems using Modified Proximal Point Algorithm in CAt(0) Spaces. i-manager’s Journal on Mathematics, 7(2), 8-16. https://doi.org/10.26634/jmat.7.2.13999

Abstract

Let (X, d) be a complete CAT(0) space and f∶ X → (-∞, ∞] be a proper convex and lower semi-continuous function. Suppose T be a nonexpansive mapping on X such that Ω=F(T) ⋂ argmin(yϵX) f(y) is nonempty. The purpose of this paper is to define a modified proximal point algorithm and prove the existence of a sequence proposed by the authors converges to Ω. In this paper, they prove strong and Δ-convergence theorems with their proposed modified proximal point algorithm in CAT(0) spaces.

Research Paper

Constrained Scalar Valued Dynamic Games and Symmetric Duality for Multi Objective Variational Problem

L. Venkateswara Reddy* , Devanandam. Dola**, B. Satyanarayana***
* Professor, Department of Information Technology, Sree Vidyanikethan Engineering College (Autonomous), Tirupati, Andhra Pradesh, India.
** Lecturer and Head, Department of Mathematics, Dharma Apparao College, Nuzivid, Krishna (Dt), Andhra Pradesh, India.
*** Professor, Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar, Andhra Pradesh, India.
Reddy. L.V., Dola. D., and Satyanarayana. B. (2018). Constrained Scalar Valued Dynamic Games and Symmetric Duality for Multi Objective Variational Problem. i-manager’s Journal on Mathematics, 7(2), 17-23. https://doi.org/10.26634/jmat.7.2.14678

Abstract

In this paper, a certain constrained scalar valued dynamic game is formulated and shown to be equivalent to a pair of multi-objective symmetric dual variational problem. These are more generation formulations then those studied easier. Further, various duality results were obtained in this context under generalized invexity conditions.

Research Paper

Influence of Viscous Dissipation on Radiative and Conducting Fluid Past an Accelerated Plate

D. Ravi Kumar* , K. Jayarami Reddy**
* Research Scholar, Department of Mathematics, Rayalaseema Univesity, Kurnool, Andhra Pradesh, India.
** Professor, Department of Mathematics, K. L. University, Vaddeswaram, Guntur, Andhra Pradesh. India.
Kumar. D.R., and Reddy. K.J. (2018). Influence of Viscous Dissipation on Radiative and Conducting Fluid Past an Accelerated Plate. i-manager’s Journal on Mathematics, 7(2), 24-31. https://doi.org/10.26634/jmat.7.2.14679

Abstract

An unsteady free convective, chemically reactive, radiation absorbing, viscous dissipative fluid past an exponentially accelerated vertical porous plate in the presence of and Soret and Dufour effects is considered. The set of nondimensional governing equations along with boundary conditions are solved numerically using finite difference method. The characteristics of pertinent physical parameters on flow quantities are examined numerically by using graphs. For the substantial interest, the fluctuations in skin friction, Nusselt number and Sherwood number are also deals through tables.

Research Paper

Thermal Diffusion and Radiation Effects on Magnetocasson Fluid Flow Past A Vertical Porous Plate

K. Venkateswara Raju* , M. C. Raju**, V. Ravi Kumar***, G. S. S. Raju****
* Assistant Professor, Department of Mathematics, Sree Vidyanikethan Engineering College (Autonomous), A. Rangampet, Tirupati, Andhra Pradesh, India.
** Coordinator Research and Development, Department of Mathematics, Annamacharya Institute of Technology and Sciences (Autonomous), Rajampet, , Andhra Pradesh, India.
*** Assistant Professor, Department of Mathematics, Annamacharya Institute of Technology and Sciences (Autonomous), Rajampet, Andhra Pradesh, India.
****The Professor and Head, Department of Mathematics, JNTU College of Engineering (Autonomous), Pulivendula, Kadapa (Dt), Andhra Pradesh, India.
Raju. K.V., Raju, M.C., Kumar. V.R., and Raju. G.S.S. (2018). Thermal Diffusion and Radiation Effects on Magnetocasson Fluid Flow Past A Vertical Porous Plate. i-manager’s Journal on Mathematics, 7(2), 32-46. https://doi.org/10.26634/jmat.7.2.14680

Abstract

In this manuscript thermal diffusion and radiation effects on magneto-Casson fluid flow past a vertical porous plate with chemical reaction is studied. The non-dimensional governing equations for momentum, energy and concentration equations have been solved by using perturbation techniques. The expressions for skin friction, Nusselt number and Sherwood number are also obtained. The effects of various physical parameters like chemical reaction parameter, Casson parameter, magnetic parameter, radiation parameter, Soret number, Schmidt number, Grashof number, Prandtl number, and modified Grashof number have been discussed in detailed through graphs and tables.