In investigating homotopy equivalences of smooth G-manifolds where G is a compact Lie group, Pétrie [3], [4], [5] makes use of proper G-maps of degree 1 from one G-module to another of the same complex dimension. The first nontrivial example of such a map, called a quasi-equivalence, was given by Pétrie [6] for two-dimensional S -modules. Necessary and sufficient conditions for the existence of...

Introduction Gromov's Polynomial Growth Theorem [Gro81] characterizes the class of virtually nilpotent groups by their asymptotic geometry. Since Gromov's theorem it has been a major open question (see, e.g. [GH91]) to find an appropriate generalization for solvable groups. This paper gives the first step in that direction. One fundamental class of examples of finitely-generated solvable groups...

Let G(v, e) be the set of all simple graphs with v vertices and e edges and let P2(G) = ∑ di denote the sum of the squares of the degrees, d1, . . . , dv, of the vertices of G. It is known that the maximum value of P2(G) for G ∈ G(v, e) occurs at one or both of two special graphs in G(v, e)—the quasi-star graph or the quasi-complete graph. For each pair (v, e), we determine which of these two g...

Let G(v, e) be the set of all simple graphs with v vertices and e edges and let P2(G) = ∑ d i denote the sum of the squares of the degrees, d1, . . . , dv, of the vertices of G. It is known that the maximum value of P2(G) for G ∈ G(v, e) occurs at one or both of two special graphs in G(v, e)—the quasi-star graph or the quasi-complete graph. For each pair (v, e), we determine which of these two ...

Let Modg be the modular group of surfaces of genus g. Each element [h] ∈ Modg induces in the integer homology of a surface of genus g a symplectic automorphism H([h]) and Poincaré shown that H : Modg → Sp(2g,Z) is an epimorphism. The theory of real algebraic curves justify the definition of real Riemann surface as a Riemann surface S with an anticonformal involution σ. Let (S, σ) be a real Riem...

We propose a new approach to superrigidity phenomena and implement it for lattice representations and measurable cocycles with Homeo+(S ) as the target group. We are motivated by Ghys’ theorem stating that any representation ̺ : Γ → Homeo+(S ) of an irreducible lattices Γ in semi-simple real Lie groups G of higher rank, either has a finite orbit or, up to a semiconjugacy, extends to G which acts...

We propose a new approach to superrigidity phenomena and implement it for lattice representations and measurable cocycles with Homeo+(S ) as the target group. We are motivated by Ghys’ theorem stating that any representation ̺ : Γ → Homeo+(S ) of an irreducible lattice Γ in a semi-simple real Lie group G of higher rank, either has a finite orbit or, up to a semi-conjugacy, extends to G which act...

A lemma of Fouquet implies that a claw-free graph contains an induced C5, contains no odd hole, or is quasi-line. In this paper we use this result to give an improved shortest-oddhole algorithm for claw-free graphs by exploiting the structural relationship between line graphs and quasi-line graphs suggested by Chudnovsky and Seymour’s structure theorem for quasi-line graphs. Our approach involv...

cyanobacteria have been considered due to producuction of types of secondary metabolites and antimicrobial compounds. this research designated for optimization of bg11 culture medium to increase the antifungal activity of cyanobacterium sp. pgu1 against the plant pathogenic fungi, fusarium oxysporum f. sp lycopersici and rhizoctonia solaniag2-2-iiib. cyanobacterium was isolated from the persian...

We say that a family of graphs G = {Gn : n ≥ 1} is p-quasi-random, 0 < p < 1, if it shares typical properties of the random graph G(n, p); for a definition, see below. We denote by Q(p) the class of all graphs H for which e(Gn) ≥ (1 + o(1))p ( n 2 ) and the number of not necessarily induced labeled copies of H in Gn is at most (1 + o(1))pn imply that G is p-quasi-random. In this note, we show t...