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

Finite-state dimension (Dai, Lathrop, Lutz, and Mayordomo (2004)) quantifiesthe information rate in an infinite sequence as measured by finite-stateautomata. In this paper, we define a relative version of finite-statedimension. The finite-state relative dimension $dim_{FS}^Y(X)$ of a sequence$X$ relative to $Y$ is the finite-state dimension of $X$ measured using theclass of finite-state gamblers with an oracle access to $Y$. We show itsmathematical robustness by equivalently characterizing this notion using therelative block entropy rate of $X$ conditioned on $Y$.  We derive inequalities relating the dimension of a sequence to the relativedimension of its subsequences along any arithmetic progression (A.P.). Theseenable us to obtain a strengthening of Wall's Theorem on the normality of A.P.subsequences of a normal number, in terms of relative dimension. In contrast tothe original theorem, this stronger version has an exact converse yielding anew characterization of normality.  We also obtain finite-state analogues of van Lambalgen's theorem on thesymmetry of relative normality.

Finite-State Relative Dimension, dimensions of A. P. subsequences and a Finite-State van Lambalgen's theorem