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Let f1(t), ..., fn(t) be independent copies of some a.s. continuous stochastic process f(t), t ∈ [0, 1], which are observed with noise. We consider the problem of nonparametric estimation of the mean function µ(t) = Ef(t) and of the covariance function ψ(t, s) = Cov{f(t), f(s)} if the noise values of each of the copies fi(t), i = 1, ..., n, are observed in some collection of generally random (in general) time points (regressors). Under wide assumptions on the time points, we construct uniformly consistent kernel estimators for the mean and covariance functions both in the case of sparse data (where the number of observations for each copy of the stochastic process is uniformly bounded) and in the case of dense data (where the number of observations at each of n series is increasing as n → ∞). In contrast to the previous studies, our kernel estimators are universal with respect to the structure of time points, which can be either fixed rather than necessarily regular, or random rather than necessarily formed of independent or weakly dependent random variables. [ABSTRACT FROM AUTHOR]

The equilibrium states and kinetics of changes in the concentrations of carbonate system components in water with different calcium hardness are analyzed and the possibility of reducing this indicator by an environmentally friendly method, aeration, is evaluated. Changes in the concentrations of dissolved carbon dioxide, calcium ions, hydrogen ions (pH), bicarbonate and carbonate ions in water depending on the partial pressure of carbon dioxide in soil and atmospheric air and the kinetics of reducing the bicarbonate-calcium hardness of water during its natural and forced aeration were traced. It is shown that water purification by a stream of atmospheric air microbubbles with an average radius of 50 µm and a flow rate of 10 l/min can reduce the hydrogen carbonate-calcium hardness from 8 mg-eq/l to 1 mg-eq/l in 100 liters of water in 20 minutes. In the paradigm of «green chemistry», this method is more rational, environmentally friendly, and economical than ion exchange or reverse osmosis, which are currently practiced to reduce hardness in decentralized water supply systems. [ABSTRACT FROM AUTHOR]

Consistent weighted least square estimators are proposed for a wide class of nonparametric regression models with random regression function, where this real-valued random function of k arguments is assumed to be continuous with probability 1. We obtain explicit upper bounds for the rate of uniform convergence in probability of the new estimators to the unobservable random regression function for both fixed or random designs. In contrast to the predecessors' results, the bounds for the convergence are insensitive to the correlation structure of the k-variate design points. As an application, we study the problem of estimating the mean and covariance functions of random fields with additive noise under dense data conditions. The theoretical results of the study are illustrated by simulation examples which show that the new estimators are more accurate in some cases than the Nadaraya–Watson ones. An example of processing real data on earthquakes in Japan in 2012–2021 is included. [ABSTRACT FROM AUTHOR]