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

Given $N$ points $X=\{x_k\}_{k=1}^N$ on the unit circle in $\mathbb{R}^2$ anda number $0\leq p \leq \infty$ we investigate the minimizers of the functional$\sum_{k, \ell =1}^N |\langle x_k, x_\ell\rangle|^p$. While it is known thateach of these minimizers is a spanning set for $\mathbb{R}^2$, less is knownabout their number as a function of $p$ and $N$ especially for relatively small$p$. In this paper we show that there is unique minimum for this functional forall $p\leq \log 3/\log 2$ and all odd $N\geq 3$. In addition, we present somenumerical results suggesting the emergence of a phase transition phenomenon forthese minimizers. More specifically, for $N\geq 3$ odd, there exists a sequenceof number of points $\log 3/\log 2=p_1< p_2< \cdots < p_N\leq 2$ so that aunique (up to some isometries) minimizer exists on each sub-intervals $(p_k,p_{k+1})$. %In addition we conjecture that $\lim_{k\to \infty}p_{2k+1}=2$.

Phase transitions for frame potentials]{Phase transitions for the minimizers of the $p^{th}$ frame potentials in $\mathbb{R}^2$