The categorical treatment of fuzzy modules presents some problems, due to the well known fact that category is not abelian, and even normal. Our aim give a representation inside generalized modules, in fact, functor category, Mod−P, which Grothendieck category. To do that, first we consider preadditive P, defined by interval P=(0,1], build torsionfree class J hereditary torsion theory finally i...

We study a competitive optimization version of α(G), the maximum size of a matching in a graph G. Players alternate adding edges of G to a matching until it becomes a maximal matching. One player (Max) wants that matching to be large; the other (Min) wants it to be small. The resulting sizes under optimal play when Max or Min starts are denoted α g(G) and α̂ ′ g(G), respectively. We show that al...

The least cardinal λ such that some (equivalently: every) compact group with weight α admits a dense, pseudocompact subgroup of cardinality λ is denoted by m(α). Clearly, m(α) ≤ 2. We show: Theorem 3.3. Among groups of cardinality γ, the group ⊕γQ serves as a “test space” for the availability of a pseudocompact group topology in this sense: If m(α) ≤ γ ≤ 2 then ⊕γQ admits a (necessarily connect...

A graph G is well-covered if all its maximal stable sets have the same size, denoted by α(G) (M. D. Plummer, 1970). If sk denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then I(G;x) = α(G) ∑ k=0 skx k is the independence polynomial of G (I. Gutman and F. Harary, 1983). J. I. Brown, K. Dilcher and R. J. Nowakowski (2000) conjectured th...

Graph shotgun assembly refers to the problem of reconstructing a graph from collection local neighborhoods. In this paper, we consider Erdős–Rényi random graphs G(n,pn), where pn=n−α for 0<α<1. We both reconstruction up isomorphism as well exact (recovering vertex labels structure). show that given distance-1 neighborhoods, G is exactly reconstructable 0<α<1 3, but not 1 2<α<1. Given distance-2...

We consider a real analytic map-germ (f, g) : (R, 0) → (R, 0). Under some conditions, we establish degree formulas for the following quantities : χ({f = α} ∩ {g = δ} ∩B ε ), χ({f = α} ∩ {g ≥ δ} ∩B ε )− χ({f = α} ∩ {g ≤ δ} ∩B ε ), where (α, δ) is a regular value of (f, g) and 0 < |(α, δ)| ¿ ε ¿ 1.

Let X be a smooth projective curve over C which admits a finite group G of automorphisms. Then G×G acts on the product surface Y = X ×X, and so for each subgroup H ≤ G×G, the quotient Z = H\Y is a normal algebraic surface. Here we shall restrict attention to the case that the subgroup H is the graph of a group automorphism α ∈ Aut(G) of G, i.e. H = ∆α = {(g, α(g)) : g ∈ G}; we propose to call t...

Let G be a graph of order n, A(G) its adjacency matrix and D(G) the diagonal degrees G. In 2017, for every α in [0, 1], Nikiforov defined Aα(G) = αD(G) + (1 − α)A(G). this paper, we investigate Aα-spectrum graphs obtained from duplicate corona operations. As an application our results, provide conditions construction some pairs non isomorphic Aα-cospectral graphs.

We study the size of the largest clique ω(G(n, α)) in a random graphG(n, α) on n vertices which has power-law degree distribution with exponent α. We show that for ‘flat’ degree sequences with α > 2 whp the largest clique in G(n, α) is of a constant size, while for the heavy tail distribution, when 0 < α < 2, ω(G(n, α)) grows as a power of n. Moreover, we show that a natural simple algorithm wh...