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The power of DNN has been successfully demonstrated on a wide variety of high-dimensional problems that cannot be solved by conventional control design methods. These successes also uncover some fundamental and pressing challenges in understanding the representability of deep neural networks for complex and high dimensional input–output relations. Towards the goal of understanding these fundamental questions, we applied an algebraic framework developed in our previous work to analyze ReLU neural network approximation of compositional functions. We prove that for Lipschitz continuous functions, ReLU neural networks have an approximation error upper bound that is a polynomial of the network’s complexity and the compositional features. If the compositional features do not increase exponentially with dimension, which is the case in many applications, the complexity of DNN has a polynomial growth. In addition to function approximations, we also establish ReLU network approximation results for the trajectories of control systems, and for a Lyapunov function that characterizes the domain of attraction. 

Abstract We present a comprehensive study of the nonproportionality of NaI(Tl) scintillation detectors within the context of dark matter search experiments. Our investigation, which integrates COSINE-100 data with supplementary$$\gamma $$ $\gamma$spectroscopy, measures light yields across diverse energy levels from full-energy$$\gamma $$ $\gamma$peaks produced by the decays of various isotopes. These$$\gamma $$ $\gamma$peaks of interest were produced by decays supported by both long and short-lived isotopes. Analyzing peaks from decays supported only by short-lived isotopes presented a unique challenge due to their limited statistics and overlapping energies, which was overcome by long-term data collection and a time-dependent analysis. A key achievement is the direct measurement of the 0.87 keV light yield, resulting from the cascade following electron capture decay of$$\mathrm {^{22}Na}$$ ${}^{22}\mathrm{Na}$from internal contamination. This measurement, previously accessible only indirectly, deepens our understanding of NaI(Tl) scintillator behavior in the region of interest for dark matter searches. This study holds substantial implications for background modeling and the interpretation of dark matter signals in NaI(Tl) experiments. 

Abstract The recent IceCube detection of TeV neutrino emission from the nearby active galaxy NGC 1068 suggests that active galactic nuclei (AGNs) could make a sizable contribution to the diffuse flux of astrophysical neutrinos. The absence of TeVγ-rays from NGC 1068 indicates neutrino production in the vicinity of the supermassive black hole, where the high radiation density leads toγ-ray attenuation. Therefore, any potential neutrino emission from similar sources is not expected to correlate with high-energyγ-rays. Disk-corona models predict neutrino emission from Seyfert galaxies to correlate with keV X-rays because they are tracers of coronal activity. Using through-going track events from the Northern Sky recorded by IceCube between 2011 and 2021, we report results from a search for individual and aggregated neutrino signals from 27 additional Seyfert galaxies that are contained in the Swift's Burst Alert Telescope AGN Spectroscopic Survey. Besides the generic single power law, we evaluate the spectra predicted by the disk-corona model assuming stochastic acceleration parameters that match the measured flux from NGC 1068. Assuming all sources to be intrinsically similar to NGC 1068, our findings constrain the collective neutrino emission from X-ray bright Seyfert galaxies in the northern sky, but, at the same time, show excesses of neutrinos that could be associated with the objects NGC 4151 and CGCG 420-015. These excesses result in a 2.7σsignificance with respect to background expectations.