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Abstract In this paper, we investigate‐multimagic squaresof order . These are magic squares that remain magic after raising each element to the th power for all . Given , we consider the problem of establishing the smallest integer for which there existnontrivial‐multimagic squares of order . Previous results on multimagic squares show that for large . We use the Hardy–Littlewood circle method to improve this to The intricate structure of the coefficient matrix poses significant technical challenges for the circle method. We overcome these obstacles by generalizing the class of Diophantine systems amenable to the circle method and demonstrating that the multimagic square system belongs to this class for all . 

Abstract We derive, via the Hardy–Littlewood method, an asymptotic formula for the number of integral zeros of a particular class of weighted quartic forms under the assumption of nonsingular local solubility. Our polynomials satisfy the condition that . Our conclusions improve on those that would follow from a direct application of the methods of Birch. For example, we show that in many circumstances the expected asymptotic formula holds when and . 

Transmission line outage detection plays an important role in maintaining the reliability of electric power systems. Most existing methods rely on optimization models to estimate the outage of transmission lines, and the process is computationally burdensome. In this study, we propose a transmission line outage detection method using machine learning. Using this method, we could monitor the power flow of one line and estimate whether another line is in service or not, despite the load fluctuations in the system. The study also investigates the principles for observation point selection and the effectiveness of this method in detecting the outage of transmission lines with different levels of power flows. The method was implemented on an IEEE 118-bus system, and results show that the method is effective for transmission lines with all levels of power flows, and line outage distribution factors (LODF) are good indicators in observation point selection. 

Abstract 2 + 2 Photocycloadditions are idealized, convergent construction approaches of 4-membered heterocyclic rings, including azetidines. However, methods of direct excitation are limited by the unfavorable photophysical properties of imines and electronically unbiased alkenes. Here, we report copper-catalyzed photocycloadditions of non-conjugated imines and alkenes to produce a variety of substituted azetidines. Design principles allow this base metal-catalyzed method to achieve 2 + 2 imine-olefin photocycloaddition via selective alkene activation through a coordination-MLCT pathway supported by combined experimental and computational mechanistic studies.