The following known selection theorem is sharpened, primarily, by weakening the hypothesis that all the sets φ(x) are closed in Y: Let X be paracompact with dimX = 0, let Y be completely metrizable and let φ:X → 𝓕(Y) be l.s.c. Then φ has a selection.
Mark Kac (1914-1984) was born in the then Russian and now Ukrainian city of Kremenec, also the birthplace of the renowned violinist Isaac Stern. In 1922, when he was eight years old, the town became part of Poland and assumed the name Krzemieniec. Because he received virtually all of his education in Polish and was a Polish citizen on arriving, in 1938, in the United States, he always considered himself to be of Polish origin, though the truth is more complicated.
Let be a stochastically continuous, separable, Gaussian process with . A sufficient condition, in terms of the monotone rearrangement of , is obtained for to have continuous sample paths almost surely. This result is applied to a wide class of random series of functions, in particular, to random Fourier series.
It is a consequence of the classification of finite simple groups that every non-abelian simple group contains a subgroup which is a minimal simple group.
A one-dimensional lattice of SIR (susceptible/infected/removed) epidemic centres is considered numerically and analytically. The limiting solutions describing the behaviour of the standard SIR model with a small number of initially infected individuals are derived, and expressions found for the duration of an outbreak. We study a model for a weakly mixed population distributed between the interacting centres. The centres are modelled as SIR nodes with interaction between sites determined by a diffusion-type...
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