A TikhonovG-space not admitting a compact HausdorffG-extension orG-linearization

Abstract

The properties of air at meteorological temperatures relevant to sound propagation and shock wave structure are reviewed. Of particular interest is the irreversible process of vibrational relaxation which describes the transfer of energy to or from the vibrational modes of the molecules and which dominates the absorption of audible sound in air. A detailed discussion of the structure and propagation of weak nonlinear waves in air shows that relaxation is again the dominant effect. As an alternative method to the usual approach of non-linear acoustics the exact gas-dynamic equations are solved numerically for a number of simple flow situations and exact results are obtained for the structure of steady waves. Estimates are obtained for the propagation distances required for the development of waveforms into steady profiles, and these distances are found under some circumstances to be greater than the dimensions of the Earth's atmosphere. The results are confirmed by laboratory experiments in CO2 and N2O and applied to waves in air with special reference to the sonic bangs of supersonic aircraft. Very recent results of applying the same alternative approach to periodic waves are reviewed briefly,

Strong local homogeneity and coset spaces

“…See Uspenskiȋ [8] for details (I am indebted to Michael Megrelishvili for informing me about this result). As was shown by Megrelishvili [3], not all actions can be 'equivariantly compactified', even if the group and the space under consideration are both Polish.…”

A topological transformation group without non-trivial equivariant compactifications

“…However, Megrelishvili [12,13] has shown it is not so, by constructing an example where both G and X are Polish, yet the embedding i is never topological for any equivariant compactification of X. More examples can be found in the work of Megrelishvili and Scarr [15] and Sokolovskaya [21].…”

“…An equivariant compactification of X is a compact space K, equipped with a continuous action by G, together with a continuous equivariant map i : X → K having a dense image in K. (See [23,24,25,22,12,14,16].) The question of interest is whether, given G and X, there exists a compactification into which X embeds topologically, that is, i is a homeomorphism onto its image.…”

The equivariant universality and couniversality of the Cantor cube

“…Some special results can be found in [8,9]. The dimension of the greatest ambit β G G may be greater than the topological dimension of G (simply take a cyclic dense subgroup G of the circle group; then dim G = 0 and dim β G G = 1).…”