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 Tools annual

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 Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

A set of positive integers <inline-formula> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> is called a Golomb ruler if the difference between two distinct elements of <inline-formula> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> are different, equivalently if the sums of two elements are different (<inline-formula> <tex-math notation="LaTeX">$B_{2}$ </tex-math></inline-formula> set, Sidon set). An extension of this concept is to consider that the sum of <inline-formula> <tex-math notation="LaTeX">$h $ </tex-math></inline-formula> elements in <inline-formula> <tex-math notation="LaTeX">$A $ </tex-math></inline-formula> are all different, except for permutation of the summands, with <inline-formula> <tex-math notation="LaTeX">$h \geq 2 $ </tex-math></inline-formula>, in this case it is said that <inline-formula> <tex-math notation="LaTeX">$A $ </tex-math></inline-formula> is a set <inline-formula> <tex-math notation="LaTeX">$B_{h} $ </tex-math></inline-formula>, the length of <inline-formula> <tex-math notation="LaTeX">$A $ </tex-math></inline-formula> is given by <inline-formula> <tex-math notation="LaTeX">$\ell (A)\,\,= \max A- \min A $ </tex-math></inline-formula>. One problem associated with this type of set is that of the optimal dense <inline-formula> <tex-math notation="LaTeX">$B_{h} $ </tex-math></inline-formula> sets, that is, determining the greatest cardinal of a set <inline-formula> <tex-math notation="LaTeX">$B_{h} $ </tex-math></inline-formula> contained in the integer interval <inline-formula> <tex-math notation="LaTeX">$\left [{1, N }\right] $ </tex-math></inline-formula>, for this defines the function <inline-formula> <tex-math notation="LaTeX">$F_{h} (N)~$ </tex-math></inline-formula>. Another problem that can be associated is the optimally short <inline-formula> <tex-math notation="LaTeX">$B_{h} $ </tex-math></inline-formula> sets, that is, finding a shorter <inline-formula> <tex-math notation="LaTeX">$B_{h} $ </tex-math></inline-formula> set with <inline-formula> <tex-math notation="LaTeX">$m $ </tex-math></inline-formula> elements, for which the <inline-formula> <tex-math notation="LaTeX">$G_{h} (m)~$ </tex-math></inline-formula> function is defined. In this paper we are going to prove that these two problems are inverse, that is, that the functions <inline-formula> <tex-math notation="LaTeX">$G_{h} (m)~$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$F_{h} (N)~$ </tex-math></inline-formula> have inverse relationships. Furthermore, the asymptotic behavior of the <inline-formula> <tex-math notation="LaTeX">$G_{h} (m)~$ </tex-math></inline-formula> function is studied, obtaining some upper and lower bounds, we also obtain tables of <inline-formula> <tex-math notation="LaTeX">$B_{3}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$B_{4}$ </tex-math></inline-formula> near-optimal up to <inline-formula> <tex-math notation="LaTeX">$m = 31$ </tex-math></inline-formula>.

Bh Sets as a Generalization of Golomb Rulers