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...

Using the notion of anti fuzzy points and its besideness to and non-quasi-coincidence with a fuzzy set, new concepts of an anti fuzzy subalgebras in BCK/BCI-algebras are introduced and their inter-relations and related properties are investigated in [3]. The notion of the new fuzzy subalgebra with thresholds are introduced and relationship between this notion and the new fuzzy subalgebra of a B...

American beech gaps in the southern Appalachian mountains are currently threatened by an insect-mediated disease complex known as beech bark disease. Cryptococcus fagisuga Lindinger, or beech scale, a major component of beech bark disease, wounds trees through feeding on vascular tissue through the outer cambium. This feeding method leaves numerous wounds and provides entryways for infection by...