A subgroup H of a given group G is said to be autopermutable, if HH(α) = H(α)H for all α ∈ Aut(G). We also call H a self-autopermutable subgroup of G, when HH(α) = H(α)H implies that H(α) = H. Moreover, G is said to be EAP-group, if every subgroup of G is autopermutable. One notes that if α runs over the inner automorphisms of the group, one obtains the notions of conjugate-permutability, self-...

be the minimal number of uniform trees with g-splitting needed to ∀ ∞-cover a uniform tree with f-splitting. c ∃ f ,g is the dual notion for the ∃ ∞-cover. Assuming CH and given ℵ 1 many (sufficiently different) pairs (f ǫ , g ǫ) and cardinals κ ǫ such that κ ℵ 0 ǫ = κ ǫ , we construct a partial order forcing that c ∃ fǫ ,gǫ = c ∀ fǫ ,gǫ = κ ǫ. For this, we introduce a countable support semipro...

We prove that for every > 0 and positive integer r, there exists ∆0 = ∆0( ) such that if ∆ > ∆0 and n > n(∆, , r) then there exists a packing of Kn with b(n − 1)/∆c graphs, each having maximum degree at most ∆ and girth at least r, where at most n edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G, f) denote ...

Let G = (V,E) be a simple and undirected graph. For some integer k > 1, a set D ⊆ V is said to be a k-dominating set in G if every vertex v of G outside D has at least k neighbors in D. Furthermore, for some real number α with 0 < α 6 1, a set D ⊆ V is called an α-dominating set in G if every vertex v of G outside D has at least α×dv neighbors in D, where dv is the degree of v in G. The cardina...

A 2-hued coloring of a graph G (also known as conditional (k, 2)-coloring and dynamic coloring) is a coloring such that for every vertex v ∈ V (G) of degree at least 2, the neighbors of v receive at least 2 colors. The smallest integer k such that G has a 2-hued coloring with k colors, is called the 2-hued chromatic number of G and denoted by χ2(G). In this paper, we will show that if G is a re...

Exercise [1.1.13] (a) Let G = ⋂ α Fα, with each Fα a σ-algebra. Since Fα a σ-algebra, we have that Ω ∈ Fα, and as this applies for all α, it follows that Ω ∈ G. Suppose now that A ∈ G. That is, A ∈ Fα for all α. Since each Fα is a σ-algebra, it follows that A ∈ Fα for all α, and hence A ∈ G. Similarly, let A = ⋃ iAi for some countable collection A1, A2, . . . of elements of G. By definition of ...

Let G be a finite solvable group, given through refined consistent polycyclic presentation, and α an automorphism of G, its images the generators G. In this paper, we discuss algorithms for computing order as well cycle length element under α. We give correctness proofs theoretical complexity these algorithms. Along way, carry out detailed analyses several classical on groups.

Tumor necrosis factor-α (TNF-α), an important factor in systematic inflammation, is reportedly involved in several cancer types. The TNF-α -308 G>A (rs1800629) polymorphism in the promoter region influences TNF-α production. The association between TNF-α -308 G>A polymorphism and colorectal cancer (CRC) is not fully understood, especially the connections between TNF-α -308 G>A polymorphism and ...

mean labelings are a type of additive vertex labeling. this labeling assigns non-negative integers to the vertices of a graph in such a way that all edge-weights are different, where the weight of an edge is defined as the mean of the end-vertex labels rounded up to the nearest integer. in this paper we focus on mean labelings of some graphs that are the result of the corona operation. in parti...

We establish the following result. Theorem. Let α : G → L(X) be a σ(X, X∗) integrable bounded group representation whose Arveson spectrum Sp(α) is scattered. Then the subspace generated by all eigenvectors of the dual representation α∗ is w∗ dense in X∗. Moreover, the σ(X, X∗) closed subalgebra Wα generated by the operators αt (t ∈ G) is semisimple. If, in addition, X does not contain any copy ...