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 define and study a metric independence notion in a homogeneous metricabstract elementary class with perturbations that is $d^p$-superstable(superstable wrt. the perturbation topology), weakly simple and has completetype spaces and we give a new example of such a class based on B. Zilber'sapproximations of Weyl algebras. We introduce a way to measure the dependenceof a tuple $a$ from a set $B$ over another set $A$. We prove basic propertiesof the notion, e.g. that $a$ is independent of $B$ over $A$ in the usual senseof homogeneous model theory if and only if the measure of dependence is$<\varepsilon$ for all $\varepsilon >0$. As an example of our measure ofdependence we show a connection between the measure and entropy in models fromquantum mechanics in which the spectrum of the observable is discrete. As anapplication, we show that weak simplicity implies a very strong form ofsimplicity and study the question of when the dependence inside a set of allrealisations of some type can be seen to arise from a pregeometry in cases whenthe type is not regular. In the end of the paper, we demonstrate our notionsand results in one more example: a class built from the $p$-adic integers.

Measuring dependence in metric abstract elementary classes with perturbations