Let G be a connected semisimple Lie group of real rank one. We denote by U(g)K the algebra of left invariant differential operators on G right invariant by K, and let Z(U(g)K) be its center. In this paper we give a sufficient condition for a differential operator P ∈ Z(U(g)K) to have a fundamental solution on G. We verify that this condition implies P C∞(G) = C∞(G). If G has a compact Cartan su...

AbstractLet W be a non-empty subset of a free group. The automorphism of a group G is said to be a marginal automorphism, if for all x in G,x^−1alpha(x) in W^*(G), where W^*(G) is the marginal subgroup of G.In this paper, we give necessary and sufficient condition for a purelynon-abelian p-group G, such that the set of all marginal automorphismsof G forms an elementary abelian p-group.

This paper considers a degree sum condition sufficient to imply the existence of k vertexdisjoint cycles in a graph G. For an integer t ≥ 1, let σt (G) be the smallest sum of degrees of t independent vertices of G. We prove that if G has order at least 7k+1 and σ4(G) ≥ 8k−3, with k ≥ 2, then G contains k vertex-disjoint cycles. We also show that the degree sum condition on σ4(G) is sharp and co...

abstractlet w be a non-empty subset of a free group. the automorphism of a group g is said to be a marginal automorphism, if for all x in g,x^−1alpha(x) in w^*(g), where w^*(g) is the marginal subgroup of g.in this paper, we give necessary and sufficient condition for a purelynon-abelian p-group g, such that the set of all marginal automorphismsof g forms an elementary abelian p-group.

for a graph $g$, let $p(g,lambda)$ denote the chromatic polynomial of $g$. two graphs $g$ and $h$ are chromatically equivalent if they share the same chromatic polynomial. a graph $g$ is chromatically unique if any graph chromatically equivalent to $g$ is isomorphic to $g$. a $k_4$-homeomorph is a subdivision of the complete graph $k_4$. in this paper, we determine a family of chromatically uni...

Let G be a finite p-group and let Aut_l(G) be the group of absolute central automorphisms of G. In this paper we give necessary and sufficient condition on G such that each absolute central automorphism of G fixes the centre element-wise. Also we classify all groups of orders p^3 and p^4 whose absolute central automorphisms fix the centre element-wise.