Hence, we see the existence of π ∈ Sym(I) with α(g) = πg for all g ∈ G. (⇐) Assume π ∈ Sym(I) exists such that α(g)(i) = (π ◦ g ◦ π−1)(i).† By definition of the symmetric group, π : I → I, i → π(i) is a bijection. Let g ∈ G and i ∈ I. Observe π(g ¦ρ i) = π(g(i)) = (π ◦ g) ◦ (π−1 ◦ π)(i) = (π ◦ g ◦ π−1) ◦ π(i) = α(g) ◦ π(i)† = g ¦ρ◦α π(i) Hence, by definition 2.10.13(a), π is G-equivariant. We a...

The independence number of a graph G, denoted by α(G), is the maximum cardinality of an independent set of vertices in G. The transversal number of G is the minimum cardinality of a set of vertices that covers all the edges of G. If G is a bipartite graph of order n, then it is easy to see that n 2 ≤ α(G) ≤ n − 1. If G has no edges, then α(G) = n = n(G). Volkmann [Australas. J. Combin. 41 (2008...

The Erdős-Hajnal conjecture says that for every graph H there exists c > 0 such that max(α(G), ω(G)) ≥ n for every H-free graph G with n vertices, and this is still open when H = C5. Until now the best bound known on max(α(G), ω(G)) for C5-free graphs was the general bound of Erdős and Hajnal, that for all H, max(α(G), ω(G)) ≥ 2 √ logn) if G is H-free. We improve this when H = C5 to max(α(G), ω...

This study examines the same-order type of G, suppose G be a finite group that α(G)=S_tt∈πe(G) and define nse(G) or α(G) is whose S_t number elements order t t∈π_e (G) π_e set G. If |G|=n in then we say α_n-group. Shin [1] showed every α_2-group nilpotent α_3-group solvable. In addition, structure such proved if an α_n-group |π(G)|≤n, conjectures researcher nonabelian simple group. proves group...

Let G be a group generated by a finite set S and equipped with the associated left-invariant word metric dG. For a Banach space X let αX(G) (respectively α # X(G)) be the supremum over all α ≥ 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f : G → X and c > 0 such that for all x, y ∈ G we have ‖ f (x) − f (y)‖ ≥ c · dG(x, y). In particular, the Hilbert compre...

Abstract Using standard Quantum Electrodynamics (QED), we have calculated the anomaly in electron orbital g-factor, which was reasonable agreement with experimental data. For second-order corrections, g-factor (gL – 1) is of order α/3π, while for approximations same order, well-known spin gS-2/2=α/2π, where α = 1/137 structure constant. Unlike spin-g anomaly, orbital-g not intrinsic, but depend...

In the current study, twin wind blades are designed, fabricated and the effect of various gap ratio (g*) at various angle of attacks (α) on a next to each other twin wind blades are examined in an open-channel wind tunnel. Aerodynamic forces and moments are determined by using three-constraint force balancer. For gap ratio of zero, the aerodynamic attributes are like those of a solitary wind bl...

We study a game played on a graph by two players, named Maximizer and Minimizer. Each round two new vertices are chosen; first Maximizer chooses a vertex u that has at least one unchosen neighbor and then Minimizer chooses a neighbor of u. This process eventually produces a maximal matching of the graph. Maximizer tries to maximize the number of edges chosen, while Minimizer tries to minimize i...