Scielo RSS <![CDATA[Proyecciones (Antofagasta)]]> vol. 33 num. 4 lang. es <![CDATA[SciELO Logo]]> <![CDATA[<b>Companions of Hermite-Hadamard Inequality for Convex Functions (II)</b>]]> Companions of Hermite-Hadamard inequalities for convex functions defined on the positive axis in the case when the integral has either the weight ψ or ¹ ,t > 0 are given. Applications for special means are provided as well. <![CDATA[<b>L(</b><b>1,1)-Labeling of Direct Product of any Path</b> <b>and Cycle</b>]]> Suppose that [n] = {0, 1, 2,...,n} is a set of non-negative integers and h,k G [n].The L (h, k)-labeling of graph G is the function l : V(G) - [n] such that |l(u) - l(v)| > h if the distance d(u,v) between u and v is 1 and |l(u) - l(v)| > k if d(u,v) = 2. Let L(V(G)) = {l(v): v G V(G)} and let p be the maximum value of L(V(G)). Then p is called Xi^-number of G if p is the least possible member of [n] such that G maintains an L(h, k) - labeling. In this paper, we establish X} - numbers of Pm X Pn and Pm X Cn graphs for all m,n > 2. <![CDATA[<b>The</b><b> upper open monophonic number of a graph</b>]]> For a connected graph G of order n,a subset S of vertices of G is a monophonic set of G if each vertex v in G lies on a x-y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is defined as the monophonic number of G, denoted by m(G). A monophonic set of cardinality m(G) is called a m-set of G.A set S of vertices of a connected graph G is an open monophonic set of G if for each vertex v in G ,either v is an extreme vertex of G and v G S,or v is an internal vertex of a x-y mono-phonic path for some x,y G S. An open monophonic set of minimum cardinality is a minimum open monophonic set and this cardinality is the open monophonic number, om(G). An open monophonic set S of vertices in a connected graph G is a minimal open monophonic .set if no proper subset of S is an open monophonic set of G.The upper open monophonic number om+ (G) is the maximum cardinality of a minimal open monophonic set of G. The upper open monophonic numbers of certain standard graphs are determined. It is proved that for a graph G of order n, om(G) = n if and only if om+(G)= n. Graphs G with om(G) = 2 are characterized. If a graph G has a minimal open monophonic set S of cardinality 3, then S is also a minimum open monophonic set of G and om(G) = 3. For any two positive integers a and b with 4 < a < b, there exists a connected graph G with om(G) = a and om+(G) = b. <![CDATA[<b>On</b><b> linear maps that preserve</b> G<b>-partial-isometries in Hilbert space</b>]]> Let Ti be a complex Hilbert space and B(TT) the algebra of all bounded linear operators on H. We give the concrete forms of surjec-tive continue unital linear maps from B(TT) onto itself that preserves G-partial-isometric operators. <![CDATA[<b>Fr</b>ó<b>chet</b><b> differentiation between Menger probabilistic normed spaces</b>]]> In this paper, we define and study Menger weakly and strongly P-convergent sequences and then Menger probabilistic continuity. We also display Frechet differentiation of nonlinear operators between Menger probabilistic normed spaces. <![CDATA[<b>Difference</b> sequence spaces in cone metric space]]> In this article we introduce the notion of difference bounded, convergent and null sequences in cone metric space. We investigate their different algebraic and topological properties. <![CDATA[<b>Subseries convergence in abstract duality pairs</b>]]> Let E, F be sets, G an Abelian topological group and b : ExF - G. Then (E, F, G) is called an abstract triple. Let w(F, E) be the weakest toplogy on F such that the maps {b(x, ·): x G E} from F into G are continuous. A subset B C F is w(F,E) sequentially conditionally compact if every sequence {yk} C B has a subsequence {y nk } such that limj; b(x, y nk) exists for every x G E. It is shown that if a formal series in E is subseries convergent in the sense that for every subsequence {x nj} there is an element x G E such that Xj=! b(x nj ,y) = b(x,y) for every y G F ,then the series Xj=! b(x nj ,y) converge uniformly for y belonging to w(F, E) sequentially conditionally compact subsets ofF. This result is used to establish Orlicz-Pettis Theorems in locall convex and function spaces. Applications are also given to Uniform Boundedness Principles and continuity results for bilinear mappings. <![CDATA[<b>Topological indices of Kragujevac trees</b>]]> We find the extremal values of the energy, the Wiener index and several vertex-degree-based topological indices over the set of Kragujevac trees with the central vertex of fixed degree.