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 perform a rigorous study of the identical sphere packing problem in$\mathbb{Z}^3$ and of phase transitions in the corresponding hard-core model.The sphere diameter $D>0$ and the fugacity $u\gg 1$ are the varying parametersof the model. We solve the sphere packing problem for values $D^2= 2, 3, 4, 5,6, 8, 9, 10, 11, 12, 2\ell^2$, $\ell\in\mathbb{N}$. For values $D^2=2, 3, 5, 8,9, 10, 12, 2\ell^2$, $\ell\in\mathbb{N}$ and $u>u^0(D)$ we establish thediagram of periodic pure phases, completely or partially. For the case$D^2=2\ell^2$, $\ell\in\mathbb{N}$ we use results from Hales' proof of Kepler'sconjecture.

Kepler's conjecture and phase transitions in the high-density hard-core model on $\mathbb{Z}^3$