Scielo RSS <![CDATA[Proyecciones (Antofagasta)]]> vol. 38 num. 2 lang. es <![CDATA[SciELO Logo]]> <![CDATA[Further results on 3-product cordial labeling]]> Abstract A mapping f : V (G) → {0, 1, 2} is called 3-product cordial labeling if |vf(i) − vf(j)| ≤ 1 and |ef(i) − ef(j)| ≤ 1 for any i, j ∈ {0, 1, 2}, where vf(i) denotes the number of vertices labeled with i, ef(i) denotes the number of edges xy with f(x)f(y) ≡ i(mod 3). A graph with 3-product cordial labeing is called 3-product cordial graph. In this paper we establish that switching of an apex vertex in closed helm, double fan, book graph K1,n × K2 and permutation graph P (K2 + mK1, I) are 3-product cordial graphs. <![CDATA[The integral sine addition law]]> Abstract In the present paper we determine, in terms of characters and additive functions, the solutions of the integral functional equation for the sine addition law (G f(xyt)dµ(t) = f(x)g(y) + g(x)f(y), x, y ∈ G, where G is a locally compact Hausdorff group and µ is a regular, compactly supported, complex-valued Borel measure on G. Some consequences of this result and an example are presented. <![CDATA[A sine type functional equation on a topological group]]> Abstract In [13] H. Stetkær obtained the complex valued solutions of the functional equation f(xyz0)f(xy−1z0) = f(x)2 − f(y)2, x, y ∈ G, where G is a topological group and z0 ∈ Z(G) (the center of G). Our main goal is first to remove this restriction and second, when G is 2-divisible and abelian, we will investigate the superstability of the above functional equation. <![CDATA[On Fuzzy Λ <sub>γ</sub> -Sets and their Applications]]> Abstract The notion of Λ-fuzzy set was introduced by M. E. EI-Shafei and A. Zakari in 2006 ((20)). We examine some basic properties of it and prove some characterization theorems for the same. The paper presents a new class of fuzzy sets called fuzzy Λγ-sets that includes the class of all fuzzy γ-open sets. It also introduces the notion of fuzzy Vγ-sets as the dual concept of fuzzy Λγ sets to study the spaces constituted by those sets and obtain a completely different structure which is called fuzzy independent Alexandorff space. A stronger form of fuzzy Λb - continuity ((2)) called fuzzy Λγ-continuity is introduced and the relationships are also established with the already existing functions accordingly. Finally, fuzzy Λγ-Generalized closed sets are defined and studied with some of their applications. <![CDATA[On the (M, D) number of a graph]]> Abstract For a connected graph G = (V, E), a monophonic set of G is a set M ⊆ V (G) such that every vertex of G is contained in a monophonic path joining some pair of vertices in M. A subset D of vertices in G is called dominating set if every vertex not in D has at least one neighbour in D. A monophonic dominating set M is both a monophonic and a dominating set. The monophonic, dominating, monophonic domination number m(G), γ(G), γm(G) respectively are the minimum cardinality of the respective sets in G. Monophonic domination number of certain classes of graphs are determined. Connected graph of order p with monophonic domination number p− 1 or p is characterised. It is shown that for every two intigers a, b ≥ 2 with 2 ≤ a ≤ b, there is a connected graph G such that γm(G) = a and γg(G) = b, where γg(G) is the geodetic domination number of a graph. <![CDATA[Further inequalities for log-convex functions related to Hermite-Hadamard result]]> Abstract Some unweighted and weighted inequalities of Hermite-Hadamard type for log-convex functions defined on real intervals are given. <![CDATA[Total domination and vertex-edge domination in tres]]> Abstract: A vertex v of a graph G = (V,E) is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a vertex-edge dominating set if every edge of E is ve-dominated by at least one vertex of S. The minimum cardinality of a vertex-edge dominating set of G is the vertex-edge domination number γve(G) . In this paper we prove (γt(T)−ℓ+1)/2 ≤ γve(T) ≤(γt(T)+ℓ−1)/2 and characterize trees attaining each of these bounds. <![CDATA[Graceful centers of graceful graphs and universal graceful graphs]]> Abstract: In this paper we define graceful center of a graceful graph. We proved any graph G which admits α-labeling has at least four graceful centers. We also defined a new strong concept of universal graceful graph. Some results on ring sum of two graphs for their graceful labeling are proved. <![CDATA[On the uniform ergodic theorem in invariant subspaces]]> Abstract: Let T be a bounded linear operator on a Banach space X into itself. In this paper, we study the uniform ergodicity of the operator T|Y when Y is a closed subspace invariant under T. We show that if T satisfies , then T is uniformly ergodic on X if and only if the restriction of T to some closed subspace Y ⊂ X, invariant under T and R[(I − T)k] ⊂ Y for some integer k ≥ 1, is uniformly ergodic. Consequently, we obtain other equivalent conditions concerning the theorem of Mbekhta and Zemànek ((9), theorem 1), also to the theorem of the Gelfand-Hille type. <![CDATA[Nonlinear elliptic equations in dimension two with potentials which can vanish at infinity]]> Abstract: We will focus on the existence of nontrivial solutions to the following nonlinear elliptic equation −∆u + V (x)u = f(u), x ∈ R2, where V is a nonnegative function which can vanish at infinity or be unbounded from above, and f have exponential growth range. The proof involves a truncation argument combined with Mountain Pass Theorem and a Trudinger-Moser type inequality. <![CDATA[Spline collocation approach to study Brachistochrone problem]]> Abstract: In this paper authors discussed a problem of quickest descent, the Brachistochrone curve. Spline collocation method is used to solve the non-linear boundary value problem. The numerical results obtained are compared with the transformation method to show effectiveness and accuracy of this method. <![CDATA[Some new Ostrowski type fractional integral inequalities for generalized relative semi-(r; m, h)-preinvex mappings via Caputo <em>k</em> -fractional derivatives]]> Abstract: In the present paper, the notion of generalized relative semi-(r; m, h)-preinvex mappings is introduced and some new integral inequalities for the left-hand side of Gauss-Jacobi type quadrature formula involving generalized relative semi-(r; m, h)-preinvex mappings are given. Moreover, some new generalizations of Ostrowski type integral inequalities to generalized relative semi-(r; m, h)-preinvex mappings that are (n + 1)-differentiable via Caputo k-fractional derivatives are established. Some applications to special means are also obtain. It is pointed out that some new special cases can be deduced from main results of the article.