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

The purpose of this short expository note is to provide an example exhibitingsome of the pathological properties of real-analytic subvarieties, where thepathology can be visualized, and the proofs use only elementary properties ofanalytic functions. We construct a compact irreducible real-analytic subvariety$S$ of ${\mathbb R}^3$ of pure dimension two such that 1) the only areal-analytic function is defined in a neighbourhood of $S$ and vanishing on$S$ is the zero function, 2) the singular set of $S$ is not a subvariety of$S$, nor is it contained in any one-dimensional subvariety of $S$, 3) thevariety $S$ contains a proper subvariety of dimension two. The example showshow a badly behaved part of a subvariety can be hidden via a secondwell-behaved component to create a subvariety of a larger set. The pathology isvisualized using several figures. Examples of these phenomena are known sincethe time of Cartan, but hard to find in the English language literature.

An example of a compact non-C-analytic real subvariety of ${\mathbb R}^3$