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 demonstrate the following uniform local definable cell decompositiontheorem in this paper.  Consider a structure $\mathcal M = (M, <,0,+, \ldots)$ elementarilyequivalent to a locally o-minimal expansion of the group of reals $(\mathbb R,<,0,+)$. Let $\{A_\lambda\}_{\lambda\in\Lambda}$ be a finite family ofdefinable subsets of $M^{m+n}$. There exist an open box $B$ in $M^n$ containingthe origin and a finite partition of definable sets $M^m \times B = X_1 \cup\ldots \cup X_k$ such that $B=(X_1)_b \cup \ldots \cup (X_k)_b$ is a definablecell decomposition of $B$ for any $b \in M^m$ and $X_i \cap A_\lambda =\emptyset$ or $X_i \subset A_\lambda$ for any $1 \leq i \leq k$ and $\lambda\in \Lambda$. Here, the notation $S_b$ denotes the fiber of a definable subset$S$ of $M^{m+n}$ at $b \in M^m$.

Uniform local definable cell decomposition for locally o-minimal expansion of the group of reals