We give a short new proof of a version of the Kruskal-Katona theorem due to Lovász. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a question of Mubayi. This in turn leads to another combinatorial proof of a stability theorem for intersecting families, which was originally obtained by Fried...

The energy content of electrostatic perturbations about homogeneous equilibria is discussed. The calculation leading to the well-known dielectric (or as it is sometimes called, the wave) energy is revisited and interpreted in light of Vlasov theory. It is argued that this quantity is deficient because resonant particles are not correctly handled. A linear integral transform is presented that so...

The problem of the rectification curves and calculation (for which we shall adopt more geometric term “flattening”) surface a rotating solid generated by curve, once rotation axis has been fixed, were treated John Wallis in two short inter-related tracts on cycloids cissoids 1659. In this work, focus parabola, an ancient reproposed 17th century, solution obtained using particular series that re...