On the forces that cable webs under tension can support and how to design cable webs to channel stresses.

In many applications of structural engineering, the following question arises: given a set of forces f 1, f 2, …, f N applied at prescribed points x 1, x 2, …, x N , under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points x 1, x 2, …, x N in the two- and three-dimensional cases. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two dimensions, we show that any such web can be replaced by one in which there are at most P elementary loops, where elementary means that the loop cannot be subdivided into subloops, and where P is the number of forces f 1, f 2, …, f N applied at points strictly within the convex hull of x 1, x 2, …, x N . In three dimensions, we show that, by slightly perturbing f 1, f 2, …, f N , there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for channelling stress in desired ways.

Field patterns: a new mathematical object.

Field patterns occur in space-time microstructures such that a disturbance propagating along a characteristic line does not evolve into a cascade of disturbances, but rather concentrates on a pattern of characteristic lines. This pattern is the field pattern. In one spatial direction plus time, the field patterns occur when the slope of the characteristics is, in a sense, commensurate with the space-time microstructure. Field patterns with different spatial shifts do not generally interact, but rather evolve as if they live in separate dimensions, as many dimensions as the number of field patterns. Alternatively one can view a collection as a multi-component potential, with as many components as the number of field patterns. Presumably, if one added a tiny nonlinear term to the wave equation one would then see interactions between these field patterns in the multi-dimensional space that one can consider them to live, or between the different field components of the multi-component potential if one views them that way. As a result of [Formula: see text]-symmetry many of the complex eigenvalues of an appropriately defined transfer matrix have unit norm and hence the corresponding eigenvectors correspond to propagating modes. There are also modes that blow up exponentially with time.