Amir Khosravi

FUSION FRAMES AND g-FRAMES IN HILBERT C*-MODULES

The notion of frame has some generalizations such as frames of subspaces, fusion frames and g-frames. In this paper, we introduce fusion frames and g-frames in Hilbert C*-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. We also generalize a perturbation result in frame theory to g-frames in Hilbert spaces. We also show that tensor product of fusion frames (g-frames) is a fusion frame (g-frame) and tensor product of resolution of identity is a resolution of identity.

A New Characterization of PSL(p, q)

Abstract Order components of a finite group are introduced in Chen [Chen, G. Y. (1996c) On Thompsons conjecture. J. Algebra 185:184–193]. It was proved that PSL(3, q), where q is an odd prime power, is uniquely determined by its order components [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002a). A characterization of PSL(3, q) where q is an odd prime power. J. Pure Appl. Algebra 170(2–3): 243–254]. Also in Iranmanesh et al. [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002b). A characterization of PSL(3, q) where q = 2 n . Acta Math. Sinica, English Ser. 18(3):463–472] and [Iranmanesh, A., Alavi, S. H. (2002). A characterization of simple groups PSL(5, q). Bull. Austral. Math. Soc. 65:211–222] it was proved that PSL(3, q) for q = 2 n and PSL(5, q) are uniquely determined by their order components. In this paper we prove that PSL(p, q) can be uniquely determined by its order components, where p is an odd prime number. A main consequence of our results is the validity of Thompsons conjecture for the groups under consideration.

A NEW CHARACTERIZATION OF ALMOST SPORADIC GROUPS

Let G be a finite group. Based on the prime graph of G, the order of G can be divided into a product of coprime positive integers. These integers are called order components of G and the set of order components is denoted by OC(G). Some non-abelian simple groups are known to be uniquely determined by their order components. In this paper we prove that almost sporadic simple groups, except Aut(J2) and Aut(McL), and the automorphism group of PSL(2, 2n) where n=2s are also uniquely determined by their order components. Also we discuss about the characterizability of Aut(PSL(2, q)). As corollaries of these results, we generalize a conjecture of J. G. Thompson and another conjecture of W. Shi and J. Bi for the groups under consideration.

A NEW CHARACTERIZATION OF SOME ALTERNATING AND SYMMETRIC GROUPS

We suppose that p = 2 α 3 β +1, where α ≥ 1, β ≥ 0, and p ≥ 7 is a prime number. Then we prove that the simple groups An ,w heren = p, p +1, or p +2, and finite groups Sn ,w heren = p, p +1, are also uniquely determined by their order components. As corollaries of these results, the validity of a conjecture of J. G. Thompson and a conjecture of Shi and Bi (1990) both on An ,w heren = p, p+1, or p +2, is obtained. Also we generalize these conjectures for the groups Sn ,w here n = p, p+1. 2000 Mathematics Subject Classification: 20D05, 20D60, 20D08.

g-FRAMES AND MODULAR RIESZ BASES IN HILBERT C*-MODULES

In this paper we introduce modular Riesz basis, modular g-Riesz basis in Hilbert C*-modules in a very natural way and we show that they share many properties with Riesz basis and g-Riesz basis in Hilbert spaces. We also found that by using the fact that every finitely or countably generated Hilbert C*-module over a unital C*-algebra has a standard Parseval frame, we characterize g-frames, modular Riesz bases and modular g-Riesz bases. Finally we obtain a perturbation result for modular g-Riesz bases.

R-duality in g-frames

Recently, the concept of g-Riesz dual sequences for g-Bessel sequences has been introduced. In this paper, we investigate under what conditions a g-Riesz sequence Φ = {Φj ∈ L(H,Hj) : j ∈ I} is the g-Riesz dual sequence of a given g-frame Λ = {Λi ∈ L(H,Hi) : i ∈ I}.