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Extremal Configurations of Hinge Structures

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 Added by Ileana Streinu
 Publication date 2008
  fields
and research's language is English




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We study body-and-hinge and panel-and-hinge chains in R^d, with two marked points: one on the first body, the other on the last. For a general chain, the squared distance between the marked points gives a Morse-Bott function on a torus configuration space. Maximal configurations, when the distance between the two marked points reaches a global maximum, have particularly simple geometrical characterizations. The three-dimensional case is relevant for applications to robotics and molecular structures.



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Motivated by the hinge structure present in protein chains and other molecular conformations, we study the singularities of certain maps associated to body-and-hinge and panel-and-hinge chains. These are sequentially articulated systems where two consecutive rigid pieces are connected by a hinge, that is, a codimension two axis. The singularities, or critical points, correspond to a dimensional drop in the linear span of the axes, regarded as points on a Grassmann variety in its Pl{u}cker embedding. These results are valid in arbitrary dimension. The three dimensional case is also relevant in robotics.
We prove configuration results for extremal Type II codes, analogous to the configuration results of Ozeki and of the second author for extremal Type II lattices. Specifically, we show that for $n in {8, 24, 32, 48, 56, 72, 96}$ every extremal Type II code of length $n$ is generated by its codewords of minimal weight. Where Ozeki and Kominers used spherical harmonics and weighted theta functions, we use discrete harmonic polynomials and harmonic weight enumerators. Along we way we introduce $tfrac12$-designs as a discrete analog of Venkovs spherical designs of the same name.
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For every positive integer $t$ we construct a finite family of triple systems ${mathcal M}_t$, determine its Tur{a}n number, and show that there are $t$ extremal ${mathcal M}_t$-free configurations that are far from each other in edit-distance. We also prove a strong stability theorem: every ${mathcal M}_t$-free triple system whose size is close to the maximum size is a subgraph of one of these $t$ extremal configurations after removing a small proportion of vertices. This is the first stability theorem for a hypergraph problem with an arbitrary (finite) number of extremal configurations. Moreover, the extremal hypergraphs have very different shadow sizes (unlike the case of the famous Turan tetrahedron conjecture). Hence a corollary of our main result is that the boundary of the feasible region of ${mathcal M}_t$ has exactly $t$ global maxima.
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