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1. On the general dyadic grids on

   https://doi.org/10.4153/S0008414X22000360  

   Anderson, Theresa C. ; Hu, Bingyang ( July 2022 , Canadian Journal of Mathematics)

   Abstract Adjacent dyadic systems are pivotal in analysis and related fields to study continuous objects via collections of dyadic ones. In our prior work (joint with Jiang, Olson, and Wei), we describe precise necessary and sufficient conditions for two dyadic systems on the real line to be adjacent. Here, we extend this work to all dimensions, which turns out to have many surprising difficulties due to the fact that $d+1$ , not $2^d$ , grids is the optimal number in an adjacent dyadic system in $\mathbb {R}^d$ . As a byproduct, we show that a collection of $d+1$ dyadic systems in $\mathbb {R}^d$ is adjacent if and only if the projection of any two of them onto any coordinate axis are adjacent on $\mathbb {R}$ . The underlying geometric structures that arise in this higher-dimensional generalization are interesting objects themselves, ripe for future study; these lead us to a compact, geometric description of our main result. We describe these structures, along with what adjacent dyadic (and n -adic, for any n ) systems look like, from a variety of contexts, relating them to previous work, as well as illustrating a specific exa. 

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2. On the bordism group for group actions on the torus

   https://doi.org/10.5802/aif.3480  

   Mann, Kathryn ; Nariman, Sam ( July 2022 , Annales de l'Institut Fourier)
3. On the trace of random walks on random graphs: ON THE TRACE OF RANDOM WALKS ON RANDOM GRAPHS

   https://doi.org/10.1112/plms.12083  

   Frieze, Alan ; Krivelevich, Michael ; Michaeli, Peleg ; Peled, Ron ( April 2018 , Proceedings of the London Mathematical Society)
4. On the Local Linear Rate of Consensus on the Stiefel Manifold

   https://doi.org/10.1109/TAC.2023.3330735  

   Chen, Shixiang ; Garcia, Alfredo ; Hong, Mingyi ; Shahrampour, Shahin ( November 2023 , IEEE Transactions on Automatic Control)
5. On the Allee effect and directed movement on the whole space

   https://doi.org/10.3934/mbe.2023347  

   Cosner, Chris ; Rodríguez, Nancy ( January 2023 , Mathematical Biosciences and Engineering)

   It is well known that relocation strategies in ecology can make the difference between extinction and persistence. We consider a reaction-advection-diffusion framework to analyze movement strategies in the context of species which are subject to a strong Allee effect. The movement strategies we consider are a combination of random Brownian motion and directed movement through the use of an environmental signal. We prove that a population can overcome the strong Allee effect when the signals are super-harmonic. In other words, an initially small population can survive in the long term if they aggregate sufficiently fast. A sharp result is provided for a specific signal that can be related to the Fokker-Planck equation for the Orstein-Uhlenbeck process. We also explore the case of pure diffusion and pure aggregation and discuss their benefits and drawbacks, making the case for a suitable combination of the two as a better strategy.

    

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