Let $G$ be a group and $H$ a subgroup of $G$‎. ‎ $H$ is said to have semi-$Pi$-property in $G$ if there is a subgroup $T$ of $G$ such that $G=HT$ and $Hcap T$ has $Pi$-property in $T$‎. ‎In this paper‎, ‎investigating on semi-$Pi$-property of subgroups‎, ‎we shall obtain some new description of finite groups‎.

The classical Stability Theorem of Erdős and Simonovits can be stated as follows. For a monotone graph property P, let t ≥ 2 be such that t + 1 = min{χ(H) : H / ∈ P}. Then any graph G∗ ∈ P on n vertices, which was obtained by removing at most (1t + o(1)) ( n 2 ) edges from the complete graph G = Kn, has edit distance o(n) to Tn(t), the Turán graph on n vertices with t parts. In this paper we ex...

A path v1, v2, . . . , vm in a graph G is degree-monotone if deg(v1) ≤ deg(v2) ≤ · · · ≤ deg(vm) where deg(vi) is the degree of vi in G. Longest degree-monotone paths have been studied in several recent papers. Here we consider the Ramsey type problem for degree monotone paths. Denote by Mk(m) the minimum number M such that for all n ≥ M , in any k-edge coloring of Kn there is some 1 ≤ j ≤ k su...

We show a surprising connexion between a property of the inf convolutions of a family of convex lower semicontinuous functions and the fact that intersections of maximal cyclically monotone graphs are the critical set of a bipotential. We then extend the results from [4] to bipotentials convex covers, generalizing the notion of a bi-implicitly convex lagrangian cover. As an application we prove...

MOTIVATION Given a large-scale biological network represented as an influence graph, in this article we investigate possible decompositions of the network aimed at highlighting specific dynamical properties. RESULTS The first decomposition we study consists in finding a maximal directed acyclic subgraph of the network, which dynamically corresponds to searching for a maximal open-loop subsyst...

and Applied Analysis 3 4 strictly η-monotone if 〈 u − v, ηx, y > 0, ∀x, y ∈ H, x / y, u ∈ M x , v ∈ M ( y ) ; 2.5 5 strongly monotone if there exists a constant r > 0 satisfying 〈 u − v, x − y ≥ r∥x − y∥2, ∀x, y ∈ H, u ∈ M x , v ∈ My; 2.6 6 strongly η-monotone if there exists a constant r > 0 satisfying 〈 u − v, ηx, y ≥ r∥x − y∥2, ∀x, y ∈ H, u ∈ M x , v ∈ My; 2.7 7 maximal monotone resp., maxim...