This article is about contingency and determination. It identifies three “inflection points”—tipping points or points of no return—in the not-so-longue durée of Soviet history: 1929, 1959, and 1989. The article thus reflects on the collectivization of agriculture and associated brutalities; the promise and limitations of Khrushchev’s reforms as well as the appeal—again, limited—of the Soviet Union to the emerging Third World; and the opportunities presented by perestroika and glasnost to reconfigure relations and purposes of production before the waves of nationalism and neoliberal market madness washed over the Soviet Union.

This paper gives a brief overview of some configurations in which high-frequency wave propagation modelled by Helmholtz equation gives rise to solutions that vary rapidly across thin layers. The configurations are grouped according to their mathematical structure and tractability and one of them concerns a famous open problem of mathematical physics.

We consider the positive zeros j″vk, k = 1, 2,…, of the second derivative of the Bessel function Jν(x). We are interested first in how many zeros there are on the interval (0,jν1), where jν1 is the smallest positive zero of Jν(x). We show that there exists a number ƛ = —0.19937078… such that and . Moreover, j″v1 decreases to 0 and j″ν2 increases to j″01 as ν increases from ƛ to 0. Further, j″vk increases in —1 < ν< ∞, for k = 3,4,… Monotonicity properties are established also for ordinates, and the slopes at the ordinates, of the points of inflection when — 1 < ν < 0.