Hyers–Ulam stability of functional equations with a square-symmetric operation

The stability of the functional equation f(x ○ y) = H(f(x), f(y)) (x, y ∈ S) is investigated, where H is a homogeneous function and ○ is a square-symmetric operation on the set S. The results presented include and generalize the classical theorem of Hyers obtained in 1941 on the stability of the Cauchy functional equation.

A model of self-thinning through local competition.

Explanations of self-thinning in plant populations have focused on plant shape and packing. A dynamic model based on the structure of local interactions successfully reproduces the pattern and can be approximated to identify key parameters and relationships. The approach generates testable new explanations for differences between species and populations, unifies self-thinning with other patterns in plant population dynamics, and indicates why organisms other than plants can follow the law.