Let $p$, $ q$ and $r$ be fixed non-negative integers. In this note, it is shown that if $R$ is left (right) $s$-unital ring satisfying $[f(x^py^q) - x^ry, x] = 0$ ($[f(x^py^q) - yx^r, x] = 0$, respectively) where $f(\lambda ) \in {\lambda }^2{\mathbb Z}[\lambda ]$, then $R$ is commutative. Moreover, commutativity of $R$ is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results.
o reveal the dynamics of short-term photosynthetic acclimation to increased irradiance, the light response of photochemical (qp), non-photochemical (q^) and Fo (qo) quenchings of chlorophyll (Chl) fluorescence and Chl and carotenoids compositíon in Norway spruce needles were monitored within three days after transfer of saplings ffom low diffuse irradiance (maximum photosynthetic photon fluence density PPFD 50 pmol m'^ s'i) to direct sun radiation (maximum PPFD 2000 pmol m‘2 s'*). Irradiance responses of fluorescence quenching coefficients revealed the occurrence of substantíal changes in partítioning of excitation energy between photochemical reactions and radiatíonless dissipation within two days. The saturating irradiance for qj,j and the capacity of non-radiatíve dissipation processes was shifted from about 450 pmol m'^ s** to 1620 pmol m-2 s'L Whereas immediately after exposure to ťull sunlight was completely reduced at 1620 pmol m'^ s'*, two days later 40 % of was stíll present in oxidized form at this irradiance. A fast pigment photobleaching at noon prevented the over-reduction of and thus it was one of the possible short-term acclimation processes. No severe photoinhibition of photosystem 2 (PS 2) photochemistry occurred within the period of investigation as can be judged from the high F^/F^ value.