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/

Zendy Plus

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

Zendy Tools

Zendy Open

Let $A=(a_{ij})$ be an $n\times n$ random matrix with i.i.d. entries suchthat $\mathbb{E} a_{11} = 0$ and $\mathbb{E} {a_{11}}^2 = 1$. We prove that forany $\delta>0$ there is $L>0$ depending only on $\delta$, and a subset$\mathcal{N}$ of $B_2^n$ of cardinality at most $\exp(\delta n)$ such that withprobability very close to one we have $$A(B_2^n)\subset \bigcup_{y\inA(\mathcal{N})}\bigl(y+L\sqrt{n}B_2^n\bigr).$$ As an application, we show thatfor some $L'>0$ and $u\in[0,1)$ depending only on the distribution law of$a_{11}$, the smallest singular value $s_n$ of the matrix $A$ satisfies$\mathbb{P}\{s_n(A)\le \varepsilon n^{-1/2}\}\le L'\varepsilon+u^n$ for all$\varepsilon>0$. The latter result generalizes a theorem of Rudelson andVershynin which was proved for random matrices with subgaussian entries.

Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries