Topology of random linkages
Topology of random linkages
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随机连接的拓扑
DOI：
10.2140/agt.2008.8.155
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发表时间：
2008
影响因子：
0.7
通讯作者：
M. Farber
中科院分区：
文献类型：
--
作者：
M. Farber
机械连杆机构构型空间的Betti数(也称为多边形空间)取决于大量的参数--连杆的杆长。受拓扑机器人、统计形状理论和分子生物学等领域应用的启发，我们将这些长度视为随机变量，并研究了当链数n趋于无穷大时，平均Betti数的渐近值。我们建立了一个令人惊讶的事实，对于一类合理充足的概率度量序列，平均Betti数的渐近值与度量的选择无关。本文的主要结果既适用于平面连杆机构，也适用于R3中的连杆机构。我们还证明了关于Betti数的高阶矩的结果。
Betti numbers of configuration spaces of mechanical linkages (known also as polygon spaces) depend on a large number of parameters ‐ the lengths of the bars of the linkage. Motivated by applications in topological robotics, statistical shape theory and molecular biology, we view these lengths as random variables and study asymptotic values of the average Betti numbers as the number of links n tends to infinity. We establish a surprising fact that for a reasonably ample class of sequences of probability measures the asymptotic values of the average Betti numbers are independent of the choice of the measure. The main results of the paper apply to planar linkages as well as for linkages in R 3 . We also prove results about higher moments of Betti numbers.