We prove that if $G$ is a graph without 3-cycles and 4-cycles, then the discrete cubical homology of trivial in dimension $d$ for all $d\ge 2$. also construct sequence $\{G_d\}$ graphs such this nontrivial 1$. Finally, we show induced by certain coverings equals ordinary singular $2$-dimensional cell complex built from $G$, although general it differs as whole.

The Hochschild and (cotriple) cyclic homologies of crossed modules of (notnecessarily-unital) associative algebras are investigated. Wodzicki’s excision theorem is extended for inclusion crossed modules in the category of crossed modules of algebras. The cyclic and cotriple cyclic homologies of crossed modules are compared in terms of long exact homology sequence, generalising the relative cycl...

We generalize results of Calderbank, Hanlon and Robinson on the representation of the symmetric group on the homology of posets of partitions with restricted block size. Calderbank, Hanlon and Robinson consider the cases of block sizes that are congruent to 0 mod d and 1 mod d for fixed d . We derive a general formula for the representation of the symmetric group on the homology of posets of pa...

Given a function f : X → R on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of f . In addition, we quantify the robustness of the homology classes under perturbations of f using well groups. After characterizing these groups, we show how to read their ranks from the same ...

We define the robustness of a level set homology class of a function f : X → R as the magnitude of a perturbation necessary to kill the class. Casting this notion into a group theoretic framework, we compute the robustness for each class, using a connection to extended persistent homology. The special case X = R 3 has ramifications in medical imaging and scientific visualization.

We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even. For every plane diagram of an oriented tangle, we construct a chain complex whose homology is invariant under Reidemeister moves. The terms of this chain complex are modules of a suita...

It is well known that during homology recognition and strand exchange the double stranded DNA (dsDNA) in DNA/RecA filaments is highly extended, but the functional role of the extension has been unclear. We present an analytical model that calculates the distribution of tension in the extended dsDNA during strand exchange. The model suggests that the binding of additional dsDNA base pairs to the...

The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures. A canonical Lie algebroid is associated to each Maurer-Cartan element. We study the geometry underlying these Maurer-Cartan elements in the light of Lie alge...

A RecA-single-stranded DNA (RecA-ssDNA) filament searches a genome for sequence homology by rapidly binding and unbinding double-stranded DNA (dsDNA) until homology is found. We demonstrate that pulling on the opposite termini (3' and 5') of one of the two DNA strands in a dsDNA molecule stabilizes the normally unstable binding of that dsDNA to non-homologous RecA-ssDNA filaments, whereas pulli...