Ripser: efficient computation of Vietoris–Rips persistence barcodes

We present an algorithm for the computation of Vietoris–Rips persistence barcodes and describe its implementation in the software Ripser. The method relies on implicit representations of the coboundary operator and the filtration order of the simplices, avoiding the explicit construction and storage of the filtration coboundary matrix. Moreover, it makes use of apparent pairs, a simple but powerful method for constructing a discrete gradient field from a total order on the simplices of a simplicial complex, which is also of independent interest. Our implementation shows substantial improvements over previous software both in time and memory usage.

Derivations and Deformations of δ-Jordan Lie Supertriple Systems

Let T be a δ-Jordan Lie supertriple system. We first introduce the notions of generalized derivations and representations of T and present some properties. Also, we study the low-dimensional cohomology and the coboundary operator of T, and then we investigate the deformations and Nijenhuis operators of T by choosing some suitable cohomologies.

Subalgebras of the Gerstenhaber algebra of differential operators

We construct a family of subalgebras of the Gerstenhaber algebra of differential operators. The subalgebras are labeled by subsets of the additive group [Formula: see text] that are closed under addition. Each subalgebra is invariant under the Hochschild coboundary operator.

A Hodge theory for Alexandrov spaces with curvature bounded from above

It is shown that the Hodge theory for metric spaces based on the Alexander Spanier coboundary operator, in the presence of a measure previously developed in [4], holds for the class of compact Alexandrov spaces with curvature bounded from above. In particular, the real cohomology of the space is isomorphic to the corresponding space of harmonic co-chains.

GENERALIZED INVERSION OF THE HOCHSCHILD COBOUNDARY OPERATOR AND DEFORMATION QUANTIZATION

Using a derivative decomposition of the Hochschild differential complex we define a generalized inverse of the Hochschild coboundary operator. It can be applied for systematic computations of star products on Poisson manifolds.

NONABELIAN GAUGE THEORIES ON NONCOMMUTATIVE SPACES

In this paper, we describe a method for obtaining the nonabelian Seiberg-Witten map for any gauge group and to any order in θ. The equations defining the Seiberg-Witten map are expressed using a coboundary operator, so that they can be solved by constructing a corresponding homotopy operator. The ambiguities, of both the gauge and covariant type, which arise in this map are manifest in our formalism.

TETRACOMPOSITION

We consider basic algebraic constructions associated with an abstract pre-opered, such as a ⌣-algebra, total composition •, pre-coboundary operator δ tribraces {·, ·, ·} and tetrabraces {·, ·, ·, ·}. By using the Gerstenhaber method, a derivation of the precoboundary operator over the tetrabraces is calculated in terms of the ⌣- multiplication and tribraces.

Smooth Formal Embeddings and the Residue Complex

Let π:X → S be a finite type morphism of noetherian schemes. A smooth formal embedding of X (over S) is a bijective closed immersion X ⊂ 𝖃 , where 𝖃 is a noetherian formal scheme, formally smooth over S. An example of such an embedding is the formal completion 𝖃 = Y/X where X ⊂ Y is an algebraic embedding. Smooth formal embeddings can be used to calculate algebraic De Rham(co)homology.Our main application is an explicit construction of the Grothendieck residue complex when S is a regular scheme. By definition the residue complex is the Cousin complex of π!OS, as in [RD]. We start with I-C. Huang's theory of pseudofunctors on modules with 0-dimensional support, which provides a graded sheaf .We then use smooth formal embeddings to obtain the coboundary operator . We exhibit a canonical isomorphism between the complex (K·x/s, δ ) and the residue complex of [RD]. When π is equidimensional of dimension n and generically smooth we show that H-nK·x/s is canonically isomorphic to to the sheaf of regular differentials of Kunz-Waldi [KW].Another issue we discuss is Grothendieck Duality on a noetherian formal scheme 1d583 . Our results on duality are used in the construction of K·x/s.