The paper investigates prime labeling of Jahangir graph Jn,m   for n ≥ 2, m ≥ 3 provided that nm is even. We discuss prime labeling of some graph operations viz. Fusion, Switching and Duplication to prove that the Fusion of two vertices v1 and vk where k is odd in a Jahangir graph Jn,m results to prime graph provided that the product nm is even and is relatively prime to k. The Fusion of two vertices vnm + 1 and vk for any k in Jn, m is prime. The switching of vk in the cycle Cnm of the Jahangir graph Jn,m  is a prime graph provided that nm+1 is a prime number and the switching of vnm+1 in Jn, m is also a prime graph .Duplicating of vk, where k is odd integer and nm + 2 is relatively prime to k,k+2 in Jn,m is a prime graph.

We give a description of a 2-torsion free Vinberg ([Formula: see text]) ring [Formula: see text]. If every nonzero root space of [Formula: see text] for [Formula: see text] is one-dimensional where [Formula: see text] is a split abelian Cartan subring of [Formula: see text] which is nil on [Formula: see text] then [Formula: see text] is a Lie ring isomorphic to [Formula: see text]. This generalizes the known result obtained by Myung for the case that [Formula: see text] is a 2-torsion free Vinberg ([Formula: see text]) ring and is power associative. We also give a condition that a Levi factor [Formula: see text] of [Formula: see text] be an ideal of [Formula: see text] when the solvable radical of [Formula: see text] is nilpotent. We apply these results for reductive case of [Formula: see text].

The strucure of the set of all non-nilpotent (-1,1) metabelian ring is studied. An additive basis of a free (-1,1) metabelian rings is constructed. It is proved that any identity in a non-nilpotent 2, 3-torsion free (-1,1) metabelian ring of degree greater than or equal to 6 is consequence of four defining identity of M where M is the metabelian (-1,1) ring.Key Words: Non-nilpotent, variety of (-1,1) rings, free metabelian rings, (-1,1) rings. The first example of solvable but not nilpotent alternative and (-1, 1) rings were constructed by Dorofeev [4], [5]. He also gave an example of a finite dimensional right alternative right nilpotent algebra which is not nilpotent. Varieties of two-step solvable nearly associative algebras were studied by many authors [2,6,7,8,9]. Thus Medvedev [9] proved that the varieties of metabelian alternative, Jordan Mal'tsev and type (-1, 1) algebras are specht. Pchelintsev [6] obtained a series results on the structure of lattices of varieties of nearly associative metabelian algebras. In this paper, we study (-1, 1) metabelian rings. They are contained in the class of algebras of type (γ, δ). In this class of ring the square of an ideal is also an ideal and hence called 2-variety. A 2, 3-torsion free ring of type (γ, δ) if satisfies the identities (x, x, x) = 0, * The project is partially supported by University Grants Commission Grants No. 42-17/2013(SR) 2000 Mathematics Subject Classification: 17A30 115Typeset by B S P M style. c Soc. Paran. de Mat. 116K. Jayalakshmi and K. Hari Babu (x, y, z) + γ(y, z, x) + δ(z, y, x) = 0, (x, y, z) − γ(x, z, y) + (1 − δ)(y, z, x) = 0, where γ 2 − δ 2 + δ − 1 = 0, and (x, y, z) = (xy)z − x(yz) is the associator of elements x, y and z. This paper includes the five sections. In sec 2 we prove that the simplest consequences of the defining relations. In sec 3 and 4, operator of the length 3 and 4 are processed. In sec 5 the function {x, y, z} = (yx)z + (zx)y is introduced, its properties are studied, and auxiliary identities necessary for constructing additive bases in free rings are proved. In sec 6, a basis of a free (-1, 1) metabelian rings is constructed and the following main results is proved.

A numerical investigation has been performed to visualize the magnetohydrodynamic natural convective heat transfer from a heated square cylinder situated within a square enclosure subjected to nonuniform temperature distributions on the left wall. The flow inside the enclosure is unsteady, incompressible, and laminar and the working fluid is micropolar fluid with constant Prandtl number (Pr = 7). The governing equations of the flow problem are the conservation of mass, energy, and linear momentum, as well as the angular momentum equations. Governing equations formulated in dimensionless velocity and pressure form has been solved by Marker and Cell method with second‐order accuracy finite difference scheme. Comprehensive verification of the utilized numerical method and mathematical model has shown a good agreement with numerical data of other authors. The results are discussed in terms of the distribution of streamlines and isotherms and surface‐averaged Nusselt number, for combinations of Rayleigh number, Ra (103–106), Vortex viscosity parameter, K (0–5), and Ha parameter (0–50). It has been shown that an increase in the vortex viscosity parameter leads to attenuation of the convective flow and heat transfer inside the cavity.

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