A complete algorithm to compute barcodes and Jordan cells is provided in this version. The paper is accepted in in the journal Discrete & Computational Geometry. arXiv admin note: text overlap with arXiv:1210.3092 by other authors
We study circle valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of real valued maps, circle valued maps enjoy a different class of invariants called Jordan cells in addition to bar codes. We establish a relation between the homology of the source space and of its relevant subspaces with these invariants and provide a new algorithm to compute these invariants from an input matrix that encodes a circle valued map on an input simplicial complex.
Deformations of circle-valued Morse functions on surfaces
设$M$是光滑连通可定向紧致曲面。用$F(M,S^1)$表示在$M$的边界上没有临界点的所有Morse函数$f:M\to S^1$的空间，并且对于$M$的每个边界分量$V$，约束$f|_{V}:V\to S^1$是常数映射或覆叠映射。为$F(M,S^1)$赋予$C^{\infty}$ -拓扑。本文对$F(M,S^1)$的连通分支进行了分类。这一结果推广了S. V. Matveev，V. V. Sharko和作者对于Morse函数在$M$边界上局部恒定的情形。
Toroidal Coordinates: Decorrelating Circular Coordinates With Lattice Reduction
Northeastern University（东北大学）；The University of Oklahoma（俄克拉荷马大学）；University of Florida（佛罗里达大学）；University of Delaware（特拉华大学）；Colorado State University（科罗拉多州立大学）；The Ohio State University（俄亥俄州立大学）；Khoury College of Computer Sciences（胡里计算机科学学院）
de Silva、Morozov和Vejdemo-Johansson的圆坐标算法将数据集与表示数据中的$1$维孔的上同调类一起作为输入;输出是从数据到捕获该孔的圆的映射，并且在适当的意义上具有最小能量。然而，当应用到几个上同调类，输出的圆值映射可以是“几何相关”，即使选择的上同调类是线性无关的。它表明，在原来的工作，相关性较低的地图，可以得到适当的整数线性组合的上同调类，与线性组合被选中的检查。在本文中，我们确定了一个正式的概念，圈值映射之间的几何相关，在黎曼流形的情况下，对应于狄利克雷形式，一个双线性形式来自狄利克雷能量。我们描述了一个系统的程序，从一组线性无关的上同调类的数据，低能量环面值映射。我们展示我们的程序与计算的例子。我们的主要算法基于计算数论中的Lenstra-Lenstra-Lovász算法。
