The notion of generalized rank in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. However, its efficient computation has not yet been studied in the literature. We show that the generalized rank over a finite interval
Let
 NSFPAR ID:
 10475617
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Discrete & Computational Geometry
 Volume:
 71
 Issue:
 2
 ISSN:
 01795376
 Format(s):
 Medium: X Size: p. 399441
 Size(s):
 ["p. 399441"]
 Sponsoring Org:
 National Science Foundation
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Abstract I of a indexed persistence module$$\textbf{Z}^2$$ ${Z}^{2}$M is equal to the generalized rank of the zigzag module that is induced on a certain path inI tracing mostly its boundary. Hence, we can compute the generalized rank ofM overI by computing the barcode of the zigzag module obtained by restricting to that path. IfM is the homology of a bifiltrationF of simplices (while accounting for multicriticality) and$$t$$ $t$I consists of points, this computation takes$$t$$ $t$ time where$$O(t^\omega )$$ $O\left({t}^{\omega}\right)$ is the exponent of matrix multiplication. We apply this result to obtain an improved algorithm for the following problem. Given a bifiltration inducing a module$$\omega \in [2,2.373)$$ $\omega \in [2,2.373)$M , determine whetherM is interval decomposable and, if so, compute all intervals supporting its indecomposable summands. 
Abstract Consider two halfspaces
and$$H_1^+$$ ${H}_{1}^{+}$ in$$H_2^+$$ ${H}_{2}^{+}$ whose bounding hyperplanes$${\mathbb {R}}^{d+1}$$ ${R}^{d+1}$ and$$H_1$$ ${H}_{1}$ are orthogonal and pass through the origin. The intersection$$H_2$$ ${H}_{2}$ is a spherical convex subset of the$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$ ${S}_{2,+}^{d}:={S}^{d}\cap {H}_{1}^{+}\cap {H}_{2}^{+}$d dimensional unit sphere , which contains a great subsphere of dimension$${\mathbb {S}}^d$$ ${S}^{d}$ and is called a spherical wedge. Choose$$d2$$ $d2$n independent random points uniformly at random on and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of$${\mathbb {S}}_{2,+}^d$$ ${S}_{2,+}^{d}$ . A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$$\log n$$ $logn$ . The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a halfsphere.$${\mathbb {S}}_{2,+}^d$$ ${S}_{2,+}^{d}$ 
Abstract It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.
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