The nerve theorem relates the topological type of a suitably nice space with the nerve of a good cover of that space. It has many variants, such as to consider acyclic covers and numerous applications in topology including applied and computational topology. The goal of this paper is to relax the notion of a good cover to an approximately good cover, or more precisely, we introduce the notion of an $\varepsilon$-acyclic cover. We use persistent homology to make this rigorous and prove tight bounds between the persistent homology of a space endowed with a function and the persistent homology of the nerve of an $\varepsilon$-acyclic cover of the space. Our approximations are stated in terms of interleaving distance between persistence modules. Using the Mayer-Vietoris spectral sequence, we prove upper bounds on the interleaving distance between the persistence module of the underlying space and the persistence module of the nerve of the cover. To prove the best possible bound, we must introduce special cases of interleavings between persistence modules called left and right interleavings. Finally, we provide examples which achieve the bound proving the lower bound and tightness of the result.
COBISS.SI-ID: 18110809
In this paper, we study the most persistent cycles which appear from a Poisson point process. We show tight upper and lower bounds.
COBISS.SI-ID: 30755623
In this paper, we give a new algorithm for predicting the movement patters of people. Using Markov chains and Monte Carlo simulation we improve prediction by taking additional time data into account.
COBISS.SI-ID: 30967335