Computation of Cubical Steenrod Squares

Krăźál, Marek; Pilarczyk, Paweł

doi:10.1007/978-3-319-39441-1_13

Cited by 8 publications

(5 citation statements)

References 17 publications

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“…We have focused on simplicial complexes since they are better known and lead to faster computations, but there are also effective constructions of Steenrod cup-i coproducts for cubical complexes (Kadeishvili 2003;Krčál and Pilarczyk 2016;Kaufmann and Medina-Mardones 2021a). Our algorithms, presented in Sect.…”

Section: Remarkmentioning

confidence: 99%

Persistence Steenrod modules

Lupo

Medina-Mardones

Tauzin³

2022

J Appl. and Comput. Topology

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It has long been envisioned that the strength of the barcode invariant of filtered cellular complexes could be increased using cohomology operations. Leveraging recent advances in the computation of Steenrod squares, we introduce a new family of computable invariants on mod 2 persistent cohomology termed $$Sq^k$$ S q k -barcodes. We present a complete algorithmic pipeline for their computation and illustrate their real-world applicability using the space of conformations of the cyclo-octane molecule.

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Section: Remarkmentioning

confidence: 99%

Persistence Steenrod modules

Lupo

Medina-Mardones

Tauzin³

2022

J Appl. and Comput. Topology

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“…Then we again apply the EM L operator to v(x) and convert it to a cubical cochain v(x) . Persistent generators w ∈ Z n−1 (X, A r ; Z) and their Steenrod images can be computed on the cubical level [39] and the cubical persistence of the secondary obstruction is done via a cubical coboundary matrix, this time in dimension n, (n + 1).…”

Section: Experiments With Formulasmentioning

confidence: 99%

Solving equations and optimization problems with uncertainty

Franek

Krčál

Wagner

2017

J Appl. and Comput. Topology

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We study the problem of detecting zeros of continuous functions that are known only up to an error bound, extending the theoretical work of Franek and Krčál (J ACM 62(4):26:1-26:19, 2015) with explicit algorithms and experiments with an implementation (https://bitbucket.org/robsatteam/rob-sat). Further, we show how to use the algorithm for approximating worst-case optima in optimization problems in which the feasible domain is defined by the zero set of a function f : X ! R n which is only known approximately. The algorithm first identifies a subdomain A where the function f is provably non-zero, a simplicial approximation f 0 : A ! S nÀ1 of f/|f|, and then verifies non-extendability of f 0 to X to certify a zero. Deciding extendability is based on computing the cohomological obstructions and their persistence. We describe an explicit algorithm for the primary and secondary obstruction, two stages of a sequence of algorithms with increasing complexity. Using elements and techniques of persistent homology, we quantify the persistence of these obstructions and hence of the robustness of zero. We provide experimental evidence that for random Gaussian fields, the primary obstruction-a much less computationally demanding test than the secondary obstruction-is typically sufficient for approximating robustness of zero.

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“…To provide researchers with effective tools for their study, ComCH implements the constructions of [18], making available for the first time chain level representations of these invariants for spaces presented simplicially or cubically. When the prime is 2, describing Steenrod operations at the chain level is classical and there are implementations for the simplicial [19] and cubical [20] cases in Sage [21] and ChainCon [22] respectively. For odd primes, we do not know of any previous implementation either in the simplicial or cubical contexts.…”

Section: Introductionmentioning

confidence: 99%

A computer algebra system for the study of commutativity up-to-coherent homotopies

Medina-Mardones¹

2021

Preprint

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The Python package ComCH is a lightweight specialized computer algebra system that provides models for well known objects, the surjection and Barratt-Eccles operads, parameterizing the product structure of algebras that are commutative in a derived sense. The primary examples of such algebras treated by ComCH are the cochain complexes of spaces, for which it provides effective constructions of Steenrod cohomology operations at all prime.

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Computation of Cubical Steenrod Squares

Cited by 8 publications

References 17 publications

Persistence Steenrod modules

Persistence Steenrod modules

Solving equations and optimization problems with uncertainty

A computer algebra system for the study of commutativity up-to-coherent homotopies

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