In this paper, πg-closed sets and πg-open sets are used to define and investigate a new class of functions called, totally πg-continuous functions. Relationships between this new class and other classes of related functions are established.

For a permutation π of the vertex set of a graph G, the graph πG is obtained from two disjoint copies G1 and G2 of G by joining each v in G1 to π(v) in G2. Hence if π = 1, then πG = K2 × G, the prism of G. Clearly, γ(G) ≤ γ(πG) ≤ 2γ(G). We study graphs for which γ(K2 × G) = 2γ(G), those for which γ(πG) = 2γ(G) for at least one permutation π of V (G) and those for which γ(πG) = 2γ(G) for each pe...

where Πg(C) is the geometric (profinite) fundamental group of C×Spec(Ks) (i.e. Πg(C) is equal to the Galois group of the maximal unramified extension of F (C)⊗Ks). This sequence induces a homomorphism ρC from GK to Out(Πg(C)) which is the group of automorphisms modulo inner automorphisms of Πg(C). It is well known that ρC is an important tool for studying C. For instance, it determines C up to ...

In [1] Burger and Mynhardt introduced the idea of universal fixers. Let G = (V,E) be a graph with n vertices and G a copy of G. For a bijective function π : V (G) → V (G), define the prism πG of G as follows: V (πG) = V (G)∪ V (G) and E(πG) = E(G)∪E(G′)∪Mπ, where Mπ = {uπ(u) | u ∈ V (G)}. Let γ(G) be the domination number of G. If γ(πG) = γ(G) for any bijective function π, then G is called a un...

Three methods haûe been examined for eûaluating the concentration of nitrogen atoms in the afterglow of a nonequilibrium, helium-stabilized, atmospheric pressure plasma. These are nitric oxide titration, absolute emission intensity of N2(B Πg) and temporal decay of the N2(B Πg) emission. To employ the second method, the rate constants for the recombination of N atoms into N2(B Πg), at different...

The paired domination number γpr(G) of a graph G is the smallest cardinality of a dominating set S of G such that 〈S〉 has a perfect matching. The generalized prisms πG of G are the graphs obtained by joining the vertices of two disjoint copies of G by |V (G)| independent edges. We provide characterizations of the following three classes of graphs: γpr(πG) = 2γpr(G) for all πG; γpr(K2 G) = 2γpr(...

Calculations are presented which use the molecular R-matrix with pseudostates (MRMPS) method to treat electron impact electron detachment and electronic excitation of the carbon dimer anion. Resonances are found above the ionisation threshold of C−2 with 1Σ+g , Πg and Πg symmetry. These are shape resonances trapped by the effect of an attractive polarisation potential competing with a repulsive...

Mynhardt has conjectured that if G is a graph such that γ(G) = γ(πG) for all generalized prisms πG then G is edgeless. The fractional analogue of this conjecture is established and proved by showing that, if G is a graph with edges, then γf (G×K2) > γf (G).

Before embarking on the proof, we first explain Theorem 1 in a bit more detail, in particular identifying the two categories that are relevant to our discussion. So let Gk denote the category whose objects are profinite group extensions of Gk by a pro-p group, and whose morphisms are outer open Gkhomomorphisms. Thus, an object of Gk is a profinite group G together with a continuous surjection π...

We present relations between growth, growth of diameters and the rate of vanishing of the spectral gap in Schreier graphs of automaton groups. In particular, we introduce a series of examples, called Hanoi Towers groups since they model the well known Hanoi Towers Problem, that illustrate some of the possible types of behavior. 1 Actions on rooted trees, automaton groups, and Hanoi Towers group...