Recall that if R is a commutative ring, then the set R× ⊂ R of invertible elements of R is naturally an abelian group under multiplication. This construction is a functor from commutative rings to abelian groups. In general, there is no obvious relation between the additive group of a ring R and the multiplicative group of units R×. However, under certain circumstances one can define a homomorp...

Many natural sounds such as speech contain concurrent amplitude and frequency modulation (AM and FM), with the FM components often in the form of directional frequency sweeps or glides. Most studies of modulation coding, however, have employed one modulation type in stationary carriers, and in cases where mixed-modulation sounds have been used, the FM component has typically been confined to an...

The Sheldon spectrum describes a remarkable regularity in aquatic ecosystems: the biomass density as a function of logarithmic body mass is approximately constant over many orders of magnitude. While size-spectrum models have explained this phenomenon for assemblages of multicellular organisms, this paper introduces a species-resolved size-spectrum model to explain the phenomenon in unicellular...