Applications of Parameterizations of Oblique Duals of a Frame Sequence

Two parameterizations of oblique duals of a given frame sequence are applied to show that the sum of a frame sequence and one of its oblique duals may not be a frame sequence, and that type II dual is locally (and globally) optimal in some sense and that the excesses of the frame sequence and its oblique duals are the same. Other observations on the duals of a frame sequence are included also.     

A Characterization of Higher Rank Symmetric Spaces Via Bounded Cohomology

Let M be complete nonpositively curved Riemannian manifold of finite volume whose fundamental group Γ does not contain a finite index subgroup which is a product of infinite groups. We show that the universal cover is a higher rank symmetric space iff is injective (and otherwise the kernel is infinite dimensional). This is the converse of a theorem of Burger–Monod. The proof uses the celebrated Rank Rigidity Theorem, as well as a new construction of quasi-homomorphisms on groups that act on CAT(0) spaces and contain rank 1 elements. Received: September 2007, Accepted: March 2008     

Moufang sets from groups of mixed type

We consider the existence of Moufang sets related to certain groups of mixed type. This way, we obtain new examples of Moufang sets and new constructions of known classes. The most interesting class of new examples is related to the Moufang quadrangles of type and to the Ree–Tits octagon over a nonperfect field, and the root groups of each member have nilpotency class three.     

On minimum rank and zero forcing sets of a graph

For a graph G on n vertices and a field F, the minimum rank of G over F, written as mr^F(G), is the smallest possible rank over all n×n symmetric matrices over F whose (i,j)th entry (for ) is nonzero whenever ij is an edge in G and is zero otherwise. The maximum nullity of G over F is M^F(G)=n-mr^F(G). The minimum rank problem of a graph G is to determine mr^F(G) (or equivalently, M^F(G)). This problem has received considerable attention over the years. In [F. Barioli, W. Barrett, S. Butler, S.M. Cioab?, D. Cvetkovi?, S.M. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovi?, H. van der Holst, K.V. Meulen, A.W. Wehe, AIM Minimum Rank-Special Graphs Work Group, Zero forcing sets and the minimum rank of graphs, Linear Algebra Appl. 428 (2008) 1628-1648], a new graph parameter Z(G), the zero forcing number, was introduced to bound M^F(G) from above. The authors posted an attractive question: What is the class of graphs G for which Z(G)=M^F(G) for some field F? This paper focuses on exploring the above question.     

Some new non-additive maximum rank distance codes

In this paper, a construction of maximum rank distance (MRD) codes as a generalization of generalized Gabidulin codes is given. The family of the resulting codes is not covered properly by additive generalized twisted Gabidulin codes, and does not cover all twisted Gabidulin codes. When the basis field has more than two elements, this family includes also non-affine MRD codes, and such codes exist for all parameters. Therefore, these codes are the first non-additive MRD codes for most of the parameters.     

A composition formula in the rank two free group

Using the fundamental group of a punctured torus, a free group of rank two, and the fact that the natural eipmorphism from onto has as kernel the group of inner automorphisms of , we describe representatives of the conjugacy classes of generating pairs of and give explicit relations between them.

     

Canonical correlation and reduction of multiple time series

It is shown that a degenerate rankd-variate stationary time series can be reduced to a full rank time series of lower dimension via an orthogonal transformationT provided that , the canonical correlation between past and future of the time series is strictly less than one. Procedures for estimation of rank of the multiple time series,T and testing =1 are outlined, the latter is related to testing the unit root hypothesis in ARMA models.