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Abstract We consider the long time behaviour of solutions to a nonlocal reaction diffusion equation that arises in the study of directed polymers in a random environment. The model is characterized by convolution with a kernel R and an L 2 inner product. In one spatial dimension, we extend a previous result of the authors (arXiv: 2002.02799 ), where only the case R = δ was considered; in particular, we show that solutions spread according to a 2 / 3 power law consistent with the KPZ scaling conjectured for directed polymers. In the special case when R = δ , we find the exact profile of the solution in the rescaled coordinates. We also consider the behaviour in higher dimensions. When the dimension is three or larger, we show that the long-time behaviour is the same as the heat equation in the sense that the solution converges to a standard Gaussian. In contrast, when the dimension is two, we construct a non-Gaussian self-similar solution. 

A swarm of unmanned aerial vehicles (UAVs) can be used for many applications, including disaster relief, search and rescue, and establishing communication networks, due to its mobility, scalability, and robustness to failure. However, a UAV swarm’s performance is typically limited by each agent’s stored energy. Recent works have considered the usage of thermals, or vertical updrafts of warm air, to address this issue. One challenge lies in a swarm of UAVs detecting and taking advantage of these thermals. Inspired by hawks, a swarm could take advantage of thermals better than individuals due to the swarm’s distributed sensing abilities. To determine which emergent behaviors increase survival time, simulation software was created to test the behavioral models of UAV gliders around thermals. For simplicity and robustness, agents operate with limited information about other agents. The UAVs’ motion was implemented as a Boids model, replicating the behavior of flocking birds through cohesion, separation, and alignment forces. Agents equipped with a modified behavioral model exhibit dynamic flocking behavior, including relative ascension-based cohesion and relative height-based separation and alignment. The simulation results show the agents flocking to thermals and improving swarm survival. These findings present a promising method to extend the flight time of autonomous UAV swarms. 

We study the one‐dimensional KPZ equation on a large torus, started at equilibrium. The main results are optimal variance bounds in the super‐relaxation regime and part of the relaxation regime.

 

We present a novel approach to deciding the validity of formulas in first-order fixpoint logic with background theories and arbitrarily nested inductive and co-inductive predicates defining least and greatest fixpoints. Our approach is constraint-based, and reduces the validity checking problem of the given first-order-fixpoint logic formula (formally, an instance in a language called µCLP) to a constraint satisfaction problem for a recently introduced predicate constraint language. Coupled with an existing sound-and-relatively-complete solver for the constraint language, this novel reduction alone already gives a sound and relatively complete method for deciding µCLP validity, but we further improve it to a novel modular primal-dual method. The key observations are (1) µCLP is closed under complement such that each (co-)inductive predicate in the original primal instance has a corresponding (co-)inductive predicate representing its complement in the dual instance obtained by taking the standard De Morgan’s dual of the primal instance, and (2) partial solutions for (co-)inductive predicates synthesized during the constraint solving process of the primal side can be used as sound upper-bounds of the corresponding (co-)inductive predicates in the dual side, and vice versa. By solving the primal and dual problems in parallel and exchanging each others’ partial solutions as sound bounds, the two processes mutually reduce each others’ solution spaces, thus enabling rapid convergence. The approach is also modular in that the bounds are synthesized and exchanged at granularity of individual (co-)inductive predicates. We demonstrate the utility of our novel fixpoint logic solving by encoding a wide variety of temporal verification problems in µCLP, including termination/non-termination, LTL, CTL, and even the full modal µ-calculus model checking of infinite state programs. The encodings exploit the modularity in both the program and the property by expressing each loops and (recursive) functions in the program and sub-formulas of the property as individual (possibly nested) (co-)inductive predicates. Together with our novel modular primal-dual µCLP solving, we obtain a novel approach to efficiently solving a wide range of temporal verification problems. 

Solar radiation‐topography interaction plays an important role in surface energy balance over the Tibetan Plateau (TP). However, the impacts of such interaction over the TP on climate locally and in the Asian regions remain unclear. This study uses the Energy Exascale Earth System Model (E3SM) to evaluate the regional and teleconnected impacts of solar radiation‐topography interaction over the TP. Land‐atmosphere coupled experiments show that topography regulates the surface energy balance, snow processes, and surface climate over the TP across seasons. Accounting for solar radiation‐topography interaction improves E3SM simulation of surface climate. The winter cold bias in air temperature decreases from −4.57 to −3.79 K, and the wet bias in summer precipitation is mitigated in southern TP. The TP's solar radiation‐topography interaction further reduces the South and East Asian summer precipitation biases. Our results demonstrate the topographic roles in regional climate over the TP and highlight its teleconnected climate impacts.