Coboundary operators for infinite frameworks

Abstract

We consider, from the point of view of operator theory, a class of infinite matrices in which the matrix entries are determined by an underlying graph structure with accompanying geometric data. This class includes the rigidity matrices of infinite bar-joint frameworks as well as the incidence matrices of infinite directed graphs. We consider the following questions: When do these matrices give rise to bounded operators? Can we compute the operator norm? When are these operators compact? And when are they bounded below?

Symbol functions for symmetric frameworks

“…In Section 3, we adopt the approach taken in [13] and introduce the more general notions of a framework (G, ϕ) for a pair of Hilbert spaces X and Y and an accompanying coboundary matrix C(G, ϕ). This convention simplifies the proofs and also allows the results to be applied in a much wider variety of settings (as demonstrated in the final section).…”

Equilibrium stresses and rigidity for infinite tensegrities and frameworks

“…The adjustments we make are natural from the perspective of functional analysis and in particular we restrict attention to countable tensegrities G(p) which have uniform structure in the sense that there is an upper bound both to the lengths of members and to the degrees of the vertices of G. This ensures that the rigidity matrix determines a bounded linear transformation from X ⊗ R d to X ⊗ R and that its transpose matrix is a bounded linear transformation in the reverse direction. For other considerations of the rigidity matrix as a bounded linear transformation see also Owen and Power [14] and Kastis, Kitson and Power [10].…”