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Gaussian processes are widely employed as versatile modelling and predictive tools in spa- tial statistics, functional data analysis, computer modelling and diverse applications of machine learning. They have been widely studied over Euclidean spaces, where they are specified using covariance functions or covariograms for modelling complex dependencies. There is a growing literature on Gaussian processes over Riemannian manifolds in order to develop richer and more flexible inferential frameworks for non-Euclidean data. While numerical approximations through graph representations have been well studied for the Mat´ern covariogram and heat kernel, the behaviour of asymptotic inference on the param- eters of the covariogram has received relatively scant attention. We focus on asymptotic behaviour for Gaussian processes constructed over compact Riemannian manifolds. Build- ing upon a recently introduced Mat´ern covariogram on a compact Riemannian manifold, we employ formal notions and conditions for the equivalence of two Mat´ern Gaussian random measures on compact manifolds to derive the parameter that is identifiable, also known as the microergodic parameter, and formally establish the consistency of the maximum like- lihood estimate and the asymptotic optimality of the best linear unbiased predictor. The circle is studied as a specific example of compact Riemannian manifolds with numerical experiments to illustrate and corroborate the theory 

We present a first search for dark-trident scattering in a neutrino beam using a dataset corresponding to$7.2\times {10}^{20}$protons on target taken with the MicroBooNE detector at Fermilab. Proton interactions in the neutrino target at the main injector produce${\pi }^0$and$\eta$mesons, which could decay into dark-matter (DM) particles mediated via a dark photon$A^{\prime }$. A convolutional neural network is trained to identify interactions of the DM particles in the liquid-argon time projection chamber (LArTPC) exploiting its imagelike reconstruction capability. In the absence of a DM signal, we provide limits at the 90% confidence level on the squared kinematic mixing parameter${ϵ}^2$as a function of the dark-photon mass in the range$10\le M_{A^{\prime }}\le 400\mathrm{MeV}$. The limits cover previously unconstrained parameter space for the production of fermion or scalar DM particles$\chi$for two benchmark models with mass ratios$M_{\chi }/M_{A^{\prime }}=0.6$and 2 and for dark fine-structure constants$0.1\le {\alpha }_D\le 1$.

Published by the American Physical Society2024