A splash experiment was carried out on a model soil–glass beads with a diameter of 425–600 μm using high‐speed cameras and sticky paper. Two different types of particles were involved in the process: droplets of water and glass beads. We argue that the result of splash of solid particles is best modeled as a stochastic point process, that is, a random number of randomly distributed points (beads) on a plane, and provide basic physical and statistical evidence that, in medium distance range (i.e., for our experiment, in the ranges of 29–64 mm), the splash may be modeled as the Poisson point process. We also argue that, in the range between 15 and 29 mm, a distribution different than Poisson is closer to reality. These two radically different types of distributions of numbers of beads in two regions reflect the fact that the solid phase of the splash involves two types of beads: those ejected in the early stage, traveling larger distances, and those ejected later, traveling shorter distances. Information on the distributions of and relations between the numbers of splashed particles in different regions may be instrumental in understanding mechanics and scale of the spread of pollutants/pathogens and plant diseases as a result of splash. Meanwhile, we describe the distributions of the total number of beads, the maximum range, and the average distance beads particles travel in a single experiment and discuss effectiveness of detection of beads by the cameras.
In this note we investigate the asymptotic behavior of the solutions of the heat equation with random, fast oscillating potentialas ε → 0+. We assume that d ≥ 1. The field {V(x), x ∈ R d } is a zero mean, stationary Gaussian random field whose covariance function is given by R(x) = R d e ip·x a( p)| p| 2−2α−d dp, where a(·) is a compactly supported, even, bounded measurable function, continuous at 0 such that a(0) > 0 and α < 1. One can show that then R(x) ∼ C|x| 2α−2 for |x| 1 where C > 0. It has been shown earlier, see e.g. Lejay (Probab Theory Relat Fields 120:255-276, 2001), that when covariance decays sufficiently fast, i.e. α < 0 and γ = 1 the solutions of Eq. 1 converge to a solution of a certain constant coefficient parabolic ("homogenized") equation. The situation changes when the decay of correlations is slower, i.e. when α ∈ (0, 1). We prove that in that case convergence holds when γ = 1 − α and the appropriate limit is no longer
We present an example of a densely defined, linear operator on the $$l^{1}$$
l
1
space with the property that each basis vector of the standard Schauder basis of $$l^{1}$$
l
1
does not belong to its domain. Our example is based on the construction of a Markov chain with all states instantaneous given by D. Blackwell in 1958. In addition, it turns out that the closure of this operator is the generator of a strongly continuous semigroup of Markov operators associated with Blackwell’s chain.
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