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

In previous work of Cristofaro-Gardiner, Frenkel, and Schlenk, the embeddingfunction $c_b(a)$ describing the problem of symplectically embedding anellipsoid $E(1,a)$ into the smallest scaling of the polydisc $P(1,b)$ wasdetermined for all integers $b \geq 2.$ As in McDuff's work,Cristofaro-Gardiner, Frenkel, and Schlenk found staircases associated with theembedding function. More recently, Usher's work shows that the appearance ofinfinite staircases as $b$ varies is hard to predict. The intricate structurefound there suggests that determining the entirety of the graph of $c_b(a)$ forall $b$ is intractable.  In contrast with this result, we show that for every polydisc $P(1,b)$ with$b>2$, there is an explicit formula for the minimum $a$ such that the embeddingproblem is determined only by volume. That is, when the ellipsoid issufficiently stretched, there is a symplectic embedding of $E(1,a)$ fullyfilling an appropriately scaled polydisc $P(\lambda,\lambda b)$. We call thisvalue of $a$ the rigid-flexible value at $b$, or the RF-value. This formula ispiecewise smooth in $b$.  To further describe the function $RF(b)$, we investigate its behavior as$b\to 1$. Frenkel and M\"{u}ller showed that the $RF$-value at $b=1$ is $7\frac{1}{32}$ and Cristofaro-Gardiner, Frenkel, and Schlenk showed that the$RF$-value at $b=2$ is $8 \frac{1}{36}.$ By exhibiting a sequence ofobstructive classes for $b = \frac{n+1}{n}$ at $a=8$, in combination with theFrenkel-M\"{u}ller result, we show that $RF$ is discontinuous at $b=1$.

The rigid-flexible value for symplectic embeddings of four-dimensional ellipsoids into polydiscs