Scielo RSS <![CDATA[Proyecciones (Antofagasta)]]> vol. 35 num. 1 lang. pt <![CDATA[SciELO Logo]]> <![CDATA[<strong>On the classification of hypersurfaces in Euclidean spaces satisfying L<sub>r</sub>H<sub>r+1</sub> = AH<sub>r+1</sub></strong>]]> In this paper, we study isometrically immersed hypersurfaces of the Euclidean space En+1 satisfying the condition LrH r+i = λHr+1 for an integer r ( 0 ≤ r ≤ n - 1), where Hr+I is the (r + 1)th mean curvature vector field on the hypersurface, Lr is the linearized operator of the first variation of the (r + 1) th mean curvature of hypersurface arising from its normal variations. Having assumed that on a hypersurface x : Mn → En+1, the vector field Hr+i be an eigenvector of the operator Lr with a constant real eigenvalue λ, we show that, Mn has to be an Lr-biharmonic, Lr-1-type, or Lr-null-2-type hypersurface. Furthermore, we study the above condition on a well-known family of hypersurfaces, named the weakly convex hypersurfaces (i.e. on which principal curvatures are nonnegative). We prove that, any weakly convex Euclidean hypersurface satisfying the condition Lr Hr+i = λ Hr+i for an integer r ( 0 ≤ r ≤ n - 1), has constant mean curvature of order (r + 1). As an interesting result, we have that, the Lr-biharmonicity condition on the weakly convex Euclidean hypersurfaces implies the r-minimality. <![CDATA[<strong>Similarity Solution of Spherical Shock Waves -Effect of Viscosity</strong>]]> In this paper we investigated self-similar Solutions for Magneto Hy-drodynamic shock waves for the equation of state of Mie-Gruneisen type. Solutions are obtained numerically and the effect of viscosity (K) and the non-idealness parameter (d) on the self-similar solutions are studied in detail. The findings confirmed that, the non-idealness parameter and the viscosity parameter have major effect on the shock strength and the flow variables. All discontinuities of the physical pa-rameters are removed by the viscosity and complete flow field depends upon the magnitude of the viscosity. The obtained results are in good agreement with the results obtained by some of the researchers. All the analysis is presented pictorially in this paper. <![CDATA[<strong>Equi independent equitable dominating sets in graphs</strong>]]> We introduce the concept of an equi independent equitable dominating set and define equi independent equitable domination number. We also investigate the graph families whose equi independent equitable domination number and equitable domination number are same. <![CDATA[<strong>Sufficient conditions for the boundedness and square integrability of Solutions of fourth-order differential equations</strong>]]> Sufficient conditions for the boundedness and square integrability of solutions and their derivatives of certain fourth order nonlin-ear differential equation are given by means of the Lyapunov’s second method. Our results obtained in this work, generalize existing results on fourth order nonlinear differential equations in the literature. For illustration, an example is also given. <![CDATA[<strong>The forcing open monophonic number of a graph</strong>]]> For a connected graph G of order n ≥ 2, and for any mínimum open monophonic set S of G, a subset T of S is called a forcing subset for S if S is the unique minimum open monophonic set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing open monophonic number of S, de-noted by f om(S), is the cardinality of a minimum forcing subset of S. The forcing open monophonic number of G, denoted by f om(G), is f om(G) = min(f om(S)), where the minimum is taken over all minimum open monophonic sets in G. The forcing open monophonic numbers of certain standard graphs are determined. It is proved that for every pair a, b of integers with 0 ≤ a ≤ b - 4 and b ≥ 5, there exists a connected graph G such that f om(G) = a and om(G) = b. It is analyzed how the addition of a pendant edge to certain standard graphs affects the forcing open monophonic number. <![CDATA[<strong>Odd harmonious labeling of some cycle related graphs</strong>]]> A graph G(p, q) is said to be odd harmonious if there exists an in-jection f : V(G)→ {0,1, 2, ..., 2q - 1} such that the induced function f * : E(G) → {1, 3, ... 2q - 1} defined by f * (uv) = f (u) + f (v) is a bijection. A graph that admits odd harmonious labeling is called odd harmonious graph. In this paper we prove that any two even cycles sharing a common vertex and a common edge are odd harmonious graphs. <![CDATA[<strong>Coupled lower and upper solution approach for the existence of Solutions of nonlinear coupled system with nonlinear coupled boundary conditions</strong>]]> <![CDATA[<strong>Sum divisor cordial graphs</strong>]]> A sum divisor cordial labeling of a graph G with vertex set V is a bijection f from V (G) to {1, 2, ..., |V (G)|} such that an edge uv is assigned the label 1 if 2 divides f (u) + f (v) and 0 otherwise, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph with a sum divisor cordial labeling is called a sum divisor cordial graph. In this paper, we prove that path, comb, star, complete bipartite, K2 + mK1, bistar, jewel, crown, flower, gear, subdivision of the star, K1,3* K1,n and square graph of Bn,n are sum divisor cordial graphs.