The minimal infinitesimal rigidity of bar-joint frameworks in the non-Euclidean spaces (R 2 , · q ), for 1 q ∞, q = 2, is characterized in terms of (2, 2)-tight graphs. Specifically, a generically placed bar-joint framework (G, p) in the plane is minimally infinitesimally rigid with respect to a non-Euclidean q norm if and only if the underlying graph G = (V, E) contains 2|V | − 2 edges and every subgraph H = (V (H), E(H)) contains at most 2|V (H)| − 2 edges.
Recent contributions on spaceability have overlooked the applicability of results on operator range subspaces of Banach spaces or Fréchet spaces. Here we consider general results on spaceability of the complement of an operator range, some of which we extend to the complement of a union of countable chains of operator ranges. Applications we give include spaceability of the non-absolutely convergent power series in the disk algebra and of the non-absolutely p-summing operators between certain pairs of Banach spaces. Another application is to ascent and descent of countably generated sets of continuous linear operators, where we establish some closed range properties of sets with finite ascent and descent.
We characterise finite and infinitesimal rigidity for bar-joint frameworks in R d with respect to polyhedral norms (i.e. norms with closed unit ball P, a convex d-dimensional polytope). Infinitesimal and continuous rigidity are shown to be equivalent for finite frameworks in R d which are well-positioned with respect to P. An edge-labelling determined by the facets of the unit ball and placement of the framework is used to characterise infinitesimal rigidity in R d in terms of monochrome spanning trees. An analogue of Laman's theorem is obtained for all polyhedral norms on R 2 .
, is shown to be almost periodically infinitesimally rigid if and only if it is strictly periodically infinitesimally rigid and the rigid unit mode (RUM) spectrum, Ω(C), is a singleton. Moreover, the almost periodic infinitesimal flexes of C are characterised in terms of a matrix-valued function, Φ C (z), on the, determined by a full rank translation symmetry group and an associated motif of joints and bars.
A simple graph G = (V, E) is 3-rigid if its generic bar-joint frameworks in R 3 are infinitesimally rigid. Block and hole graphs are derived from triangulated spheres by the removal of edges and the addition of minimally rigid subgraphs, known as blocks, in some of the resulting holes. Combinatorial characterisations of minimal 3-rigidity are obtained for these graphs in the case of a single block and finitely many holes or a single hole and finitely many blocks. These results confirm a conjecture of Whiteley from 1988 and special cases of a stronger conjecture of Finbow-Singh and Whiteley from 2013.2010 Mathematics Subject Classification. Primary 52C25 Secondary 05C75.
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