2-Point set domination in separable graphs

In a graph [Formula: see text], a set [Formula: see text] is a [Formula: see text]-point set dominating set (in short 2-psd set) of [Formula: see text] if for every subset [Formula: see text] there exists a nonempty subset [Formula: see text] containing at most two vertices such that the induced subgraph [Formula: see text] is connected in [Formula: see text]. The [Formula: see text]-point set domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of a 2-psd set of [Formula: see text]. The main focus of this paper is to find the value of [Formula: see text] for a separable graph and thereafter computing [Formula: see text] for some well-known classes of separable graphs. Further we classify the set of all 2-psd sets of a separable graph into six disjoint classes and study the existence of minimum 2-psd sets in each class.

Magnifying Elements in Semigroups of Fixed Point Set Transformations Restricted by an Equivalence Relation

Let X be a nonempty set and ρ be an equivalence relation on X . For a nonempty subset S of X , we denote the semigroup of transformations restricted by an equivalence relation ρ fixing S pointwise by E F S X , ρ . In this paper, magnifying elements in E F S X , ρ will be investigated. Moreover, we will give the necessary and sufficient conditions for elements in E F S X , ρ to be right or left magnifying elements.

On the Connected Safe Number of Some Classes of Graphs

For a connected simple graph G , a nonempty subset S of V G is a connected safe set if the induced subgraph G S is connected and the inequality S ≥ D satisfies for each connected component D of G∖S whenever an edge of G exists between S and D . A connected safe set of a connected graph G with minimum cardinality is called the minimum connected safe set and that minimum cardinality is called the connected safe numbers. We study connected safe sets with minimal cardinality of the ladder, sunlet, and wheel graphs.

A Note on Interior Bases of Semigroups

The concept of two-sided bases of a semigroup S was introduced by Fabrici in 2009. In this paper, we introduce the concept of interior bases of a semigroup S which is based on the result of interior ideals generated by a nonempty subset of the semigroup S. Then, we study some results of a semigroup S containing interior bases and characterize when a nonempty subset of a semigroup S is an interior base of S

Prime i-Ideals in Ordered n-ary Semigroups

We study the concept of i-ideal of an ordered n-ary semigroup and give a construction of the i-ideal of an ordered n-ary semigroup generated by its nonempty subset. Moreover, we study the notions of prime, weakly prime, semiprime and weakly semiprime ideals of an ordered n-ary semigroup.

Properties of the Global Total k-Domination Number

A nonempty subset D⊂V of vertices of a graph G=(V,E) is a dominating set if every vertex of this graph is adjacent to at least one vertex from this set except the vertices which belong to this set itself. D⊆V is a total k-dominating set if there are at least k vertices in set D adjacent to every vertex v∈V, and it is a global total k-dominating set if D is a total k-dominating set of both G and G¯. The global total k-domination number of G, denoted by γktg(G), is the minimum cardinality of a global total k-dominating set of G, GTkD-set. Here we derive upper and lower bounds of γktg(G), and develop a method that generates a GTkD-set from a GT(k−1)D-set for the successively increasing values of k. Based on this method, we establish a relationship between γ(k−1)tg(G) and γktg(G), which, in turn, provides another upper bound on γktg(G).

Orthogonality in smooth countably normed spaces

We generalize the concepts of normalized duality mapping, J-orthogonality and Birkhoff orthogonality from normed spaces to smooth countably normed spaces. We give some basic properties of J-orthogonality in smooth countably normed spaces and show a relation between J-orthogonality and metric projection on smooth uniformly convex complete countably normed spaces. Moreover, we define the J-dual cone and J-orthogonal complement on a nonempty subset S of a smooth countably normed space and prove some basic results about the J-dual cone and the J-orthogonal complement of S.

On the Set Version of Selectively Star-CCC Spaces

A space X is said to be set selectively star-ccc if for each nonempty subset B of X , for each collection U of open sets in X such that B ¯ ⊂ ∪ U , and for each sequence A n : n ∈ ℕ of maximal cellular open families in X , there is a sequence A n : n ∈ ℕ such that, for each n ∈ ℕ , A n ∈ A n and B ⊂ St ∪ n ∈ ℕ A n , U . In this paper, we introduce set selectively star-ccc spaces and investigate the relationship between set selectively star-ccc and other related spaces. We also study the topological properties of set selectively star-ccc spaces. Some open problems are posed.

Invariant Means, Complementary Averages of Means, and a Characterization of the Beta-Type Means

We prove that whenever the selfmapping (M1,…,Mp):Ip→Ip, (p∈N and Mi-s are p-variable means on the interval I) is invariant with respect to some continuous and strictly monotone mean K:Ip→I then for every nonempty subset S⊆{1,…,p} there exists a uniquely determined mean KS:Ip→I such that the mean-type mapping (N1,…,Np):Ip→Ip is K-invariant, where Ni:=KS for i∈S and Ni:=Mi otherwise. Moreover min(Mi:i∈S)≤KS≤max(Mi:i∈S). Later we use this result to: (1) construct a broad family of K-invariant mean-type mappings, (2) solve functional equations of invariant-type, and (3) characterize Beta-type means.

Approximating Fixed Points of Operators Satisfying (RCSC) Condition in Banach Spaces

Let K be a nonempty subset of a Banach space E. A mapping T:K→K is said to satisfy (RCSC) condition if each a,b∈K, 1/2a−Fa≤a−b⇒Fa−Fb≤1/3a−b+a−Fb+b−Fa. In this paper, we study, under some appropriate conditions, weak and strong convergence for this class of maps through M iterates in uniformly convex Banach space. We also present a new example of mappings with condition (RCSC). We connect M iteration and other well-known processes with this example to show the numerical efficiency of our results. The presented results improve and extend the corresponding results of the literature.