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  1. We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with weights characterized by an underlying graphon. A law of large numbers result is established as the system size increases and the underlying graphons converge. The limit is given by a graphon mean field system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Well-posedness, continuity and stability of such systems are provided. We also consider a not-so-dense analogue of the finite particle system, obtained by percolation with vanishing rates and suitable scaling of interactions. A law of large numbers result is proved for the convergence of such systems to the corresponding graphon mean field system. 
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  2. Impurities in water affect ice adhesion strength on surfaces. Depending on the freezing rate, they can be trapped in ice or pushed out, forming a lubricating layer. They also affect the quasi-liquid layer between ice and surface, impacting adhesion.

     
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  3. A graph $G$ percolates in the $K_{r,s}$-bootstrap process if we can add all missing edges of $G$ in some order such that each edge creates a new copy of $K_{r,s}$, where $K_{r,s}$ is the complete bipartite graph. We study $K_{r,s}$-bootstrap percolation on the Erdős-Rényi random graph, and determine the percolation threshold for balanced $K_{r,s}$ up to a logarithmic factor. This partially answers a question raised by Balogh, Bollobás, and Morris. We also establish a general lower bound of the percolation threshold for all $K_{r,s}$, with $r\geq s \geq 3$. 
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  4. null (Ed.)
  5. Abstract

    Atomically thin membranes comprising nanopores in a 2D material promise to surpass the performance of polymeric membranes in several critical applications, including water purification, chemical and gas separations, and energy harvesting. However, fabrication of membranes with precise pore size distributions that provide exceptionally high selectivity and permeance in a scalable framework remains an outstanding challenge. Circumventing these constraints, here, a platform technology is developed that harnesses the ability of oppositely charged polyelectrolytes to self‐assemble preferentially across larger, relatively leaky atomically thin nanopores by exploiting the lower steric hindrance of such larger pores to molecular interactions across the pores. By selectively tightening the pore size distribution in this manner, self‐assembly of oppositely charged polyelectrolytes simultaneously introduced on opposite sides of nanoporous graphene membranes is demonstrated to discriminate between nanopores to seal non‐selective transport channels, while minimally compromising smaller, water‐selective pores, thereby remarkably attenuating solute leakage. This improved membrane selectivity enables desalination across centimeter‐scale nanoporous graphene with 99.7% and >90% rejection of MgSO4and NaCl, respectively, under forward osmosis. These findings provide a versatile strategy to augment the performance of nanoporous atomically thin membranes and present intriguing possibilities of controlling reactions across 2D materials via exclusive exploitation of pore size‐dependent intermolecular interactions.

     
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