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A stochastic optimization algorithm for analyzing planar central and balanced configurations in the n-body problem is presented. We find a comprehensive list of equal mass central configurations satisfying the Morse equality up to n = 12. We show some exemplary balanced configurations in the case n = 5, as well as some balanced configurations without any axis of symmetry in the cases n = 4 and n = 10.

In this paper, we present neural network methods for predicting uncertainty in atmospheric remote sensing. These include methods for solving the direct and the inverse problem in a Bayesian framework. In the first case, a method based on a neural network for simulating the radiative transfer model and a Bayesian approach for solving the inverse problem is proposed. In the second case, (i) a neural network, in which the output is the convolution of the output for a noise-free input with the input noise distribution; and (ii) a Bayesian deep learning framework that predicts input aleatoric and model uncertainties, are designed. In addition, a neural network that uses assumed density filtering and interval arithmetic to compute uncertainty is employed for testing purposes. The accuracy and the precision of the methods are analyzed by considering the retrieval of cloud parameters from radiances measured by the Earth Polychromatic Imaging Camera (EPIC) onboard the Deep Space Climate Observatory (DSCOVR).

Starting with a orientable compact real-analytic Riemannian manifold (L, g) with χ(L) = 0, we show that a small neighbourhood Op(L) of the zero section in the cotangent bundle T * L carries a Calabi-Yau structure such that the zero section is an isometrically embedded special Lagrangian submanifold.