Regular Article
On Traces ofd-stresses in the Skeletons of Lower Dimensions of Piecewise-lineard-manifolds

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Abstract

We show how a d -stress on a piecewise-linear realization of an oriented (non-simplicial, in general)d -manifold in Rdnaturally induces stresses of lower dimensions on this manifold, and discuss implications of this construction to the analysis of self-stresses in spatial frameworks. The mappings we construct are not linear, but polynomial. In the 1860–70s J. C. Maxwell described an interesting relationship between self-stresses in planar frameworks and vertical projections of polyhedral 2-surfaces. We offer a spatial analog of Maxwell’s correspondence based on our polynomial mappings. By applying our main result we derive a class of three-dimensional spider webs similar to the two-dimensional spider webs described by Maxwell. We also conjecture an important property of our mappings that is based on the lower bound theorem (g2(d +  1)  = dim Stress2   0) for d -pseudomanifolds generically realized in Rd + 1.

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1

[email protected]

2

Present address: Department of Mathematics, Cornell University.

3

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4

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