We elucidate the relationship between threshold and expectation-threshold of a down-set. Qualitatively, our main result demonstrates that there exist down-sets with polynomial gaps their thresholds expectation-thresholds; in particular, logarithmic gap predictions Kahn–Kalai Talagrand (recently proved by Park–Pham Frankston–Kahn–Narayanan–Park) about up-sets do not apply to down-sets. Quantitat...

This study presents a new method based on the combination of cryptography and information hiding methods. Firstly, the image is encoded by the Double Random Phase Encoding (DRPE) technique. The real and imaginary parts of the encoded image are subsequently embedded into an enlarged normalized host image. DRPE demands two random phase mask keys to decode the decrypted image at the destination. T...

‎for a set of non-negative integers~$l$‎, ‎the $l$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $a_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|a_u cap a_v|in l$‎. ‎the bipartite $l$-intersection number is defined similarly when the conditions are considered only for the ver...

fuzzy rough n-ary subhypergroups are introduced and characterized.