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 introduce and study natural derivatives for Christoffel and finitestandard words, as well as for characteristic Sturmian words. Thesederivatives, which are realized as inverse images under suitable morphisms,preserve the aforementioned classes of words. In the case of Christoffel words,the morphisms involved map $a$ to $a^{k+1}b$ (resp.,~$ab^{k}$) and $b$ to$a^{k}b$ (resp.,~$ab^{k+1}$) for a suitable $k>0$. As long as derivatives arelonger than one letter, higher-order derivatives are naturally obtained. Wedefine the depth of a Christoffel or standard word as the smallest order forwhich the derivative is a single letter. We give several combinatorial andarithmetic descriptions of the depth, and (tight) lower and upper bounds forit.

On Christoffel and standard words and their derivatives