English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We introduce an algorithm that constructs a discrete gradient field on anysimplicial complex. We show that, in all situations, the gradient field ismaximal possible and, in a number of cases, optimal. We make a thoroughanalysis of the resulting gradient field in the case of Munkres' discrete modelfor $\text{Conf}(K_m,2)$, the configuration space of ordered pairs ofnon-colliding particles on the complete graph $K_m$ on $m$ vertices. Togetherwith the use of Forman's discrete Morse theory, this allows us to describe infull the cohomology $R$-algebra $H^*(\text{Conf}(K_m,2);R)$ for any commutativeunital ring $R$. As an application we prove that, although $\text{Conf}(K_m,2)$is outside the "stable" regime, all its topological complexities are maximalpossible when $m\geq4$.

An algorithmic discrete gradient field and the cohomology algebra of configuration spaces of two points on complete graphs