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 Tools annual

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 Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We define the translation-invariant Laplacian $-\triangle_{\mathbb R^\infty}$on the product measurable space $\mathbb R^\infty$ as a non-negativeself-adjoint operator in some Hilbert space $L^2(\mathbb R^\infty)$, which is asubset of the set $CM(\mathbb R^\infty)$ of all complex measures on $\mathbbR^\infty$. Furthermore, we show that for any $f\in L^2(\mathbb R^n)$ and any$u\in L^2(\mathbb R^\infty)$, $e^{\sqrt{-1}\triangle_{\mathbbR^\infty}t}(f\otimes u)=(e^{\sqrt{-1}\triangle_{\mathbbR^n}t}f)\otimes(e^{\sqrt{-1}\triangle_{\mathbb R^\infty}t}u) \ (t\in(-\infty,+\infty))$ and $e^{\triangle_{\mathbb R^\infty}t}(f\otimesu)=(e^{\triangle_{\mathbb R^n}t}f)\otimes(e^{\triangle_{\mathbb R^\infty}t}u) \(t\in [0,+\infty))$ hold.

Self-adjoint Laplace operator with translation invariance on infinite-dimensional space $\mathbb R^\infty$