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

This paper considers a problem that relates to the theories of coveringarrays, permutation patterns, Vapnik-Chervonenkis (VC) classes, and probabilitythresholds. Specifically, we want to find the number of subsets of[n]:={1,2,....,n} we need to randomly select, in a certain probability space,so as to respectively "shatter" all t-subsets of [n]. Moving from subsets towords, we ask for the number of n-letter words on a q-letter alphabet that areneeded to shatter all t-subwords of the q^n words of length n. Finally, weexplore the number of random permutations of [n] needed to shatter(specializing to t=3), all length 3 permutation patterns in specifiedpositions. We uncover a very sharp zero-one probability threshold for theemergence of such shattering; Talagrand's isoperimetric inequality in productspaces is used as a key tool.

Shattering Thresholds for Random Systems of Sets, Words, and Permutations