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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
1999
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/math/9902160 |
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| _version_ | 1866911114948771840 |
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| author | Erdahl, Robert M. Rybnikov, Konstantine Ryshkov, Sergeĭ S. |
| author_facet | Erdahl, Robert M. Rybnikov, Konstantine Ryshkov, Sergeĭ S. |
| contents | We show how a $d$-stress on a piecewise-linear realization of an oriented (non-simplicial, in general) $d$-manifold in \rd naturally induces stresses of lower dimensions on this manifold, and discuss implications of this construction to the analysis of self-stresses in spatial frameworks. The constructed mappings are not linear, but polynomial. In 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 partial analog of Maxwell correspondence for self-stresses in spatial frameworks and vertical projections of 3-dimensional surfaces based on our construction of polynomial mappings. Applying this theorem we derive a class of three-dimensional spider webs similar to the family of two-dimensional spider webs described by Maxwell. In addition, we conjecture an important property of our mappings which is supported by a heuristic count based on the lower bound theorem ($g_2(d+1)=dim\:Stress_2 \ge 0$) for $d$-pseudomanifolds generically realized in ${\R}^{d+1}$ (Fogelsanger). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_9902160 |
| institution | arXiv |
| publishDate | 1999 |
| record_format | arxiv |
| spellingShingle | On traces of $d$-stresses in the skeletons of lower dimensions of homology $d$-manifolds Erdahl, Robert M. Rybnikov, Konstantine Ryshkov, Sergeĭ S. Metric Geometry We show how a $d$-stress on a piecewise-linear realization of an oriented (non-simplicial, in general) $d$-manifold in \rd naturally induces stresses of lower dimensions on this manifold, and discuss implications of this construction to the analysis of self-stresses in spatial frameworks. The constructed mappings are not linear, but polynomial. In 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 partial analog of Maxwell correspondence for self-stresses in spatial frameworks and vertical projections of 3-dimensional surfaces based on our construction of polynomial mappings. Applying this theorem we derive a class of three-dimensional spider webs similar to the family of two-dimensional spider webs described by Maxwell. In addition, we conjecture an important property of our mappings which is supported by a heuristic count based on the lower bound theorem ($g_2(d+1)=dim\:Stress_2 \ge 0$) for $d$-pseudomanifolds generically realized in ${\R}^{d+1}$ (Fogelsanger). |
| title | On traces of $d$-stresses in the skeletons of lower dimensions of homology $d$-manifolds |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/math/9902160 |