Humoral antibodies to heat-stable antigens of Renibacterium salmoninarum (Rs) were detected by enzyme-linked immunosorbent assay (ELISA) in rainbow trout Oncorhynchus mykiss and in Atlantic salmon Salmo salar challenged by immersion. A slow antibody response was found: 3% (1/30) was positive 4 wk after immersion and 72% (26/36) was positive after 8 wk. All 30 fish sampled after 4 wk were found ...

We determine necessary conditions for when powers corresponding to positive/negative coefficients of Φn are in arithmetic progression. When n = pq any primes q>p>2, our also sufficient. Finally, we generalize the result so-called inclusion-exclusion polynomials first introduced by Bachman.

Given an integer r ≥ 2 and a number υ ∈ (0, 1], consider the collection of all subsets of Z/rZ having at least υr elements. Among the sets in this collection, suppose S is any one having the minimal number of three-term arithmetic progressions, where in our terminology a three-term arithmetic progression is a triple (x, y, z) ∈ S3 satisfying x + y ≡ 2z (mod r). Note that this includes trivial p...