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We survey the theory of totally symmetric sets, with applications to homo- morphisms of symmetric groups, braid groups, linear groups, and mapping class groups. 

Following exocytosis at active zones, synaptic vesicle membranes and membrane-bound proteins must be recycled. The endocytic machinery that drives this recycling accumulates in the periactive zone (PAZ), a region of the synapse adjacent to active zones, but the organization of this machinery within the PAZ, and how PAZ composition relates to active zone release properties, remains unknown. The PAZ is also enriched for cell adhesion proteins, but their function at these sites is poorly understood. Here, using Airyscan and stimulated emission depletion imaging of Drosophila synapses, we develop a quantitative framework describing the organization and ultrastructure of the PAZ. Different endocytic proteins localize to distinct regions of the PAZ, suggesting that subdomains are specialized for distinct biochemical activities, stages of membrane remodeling, or synaptic functions. We find that the accumulation and distribution of endocytic but not adhesion PAZ proteins correlate with the abundance of the scaffolding protein Bruchpilot at active zones—a structural correlate of release probability. These data suggest that endocytic and exocytic activities are spatially correlated. Taken together, our results identify novel relationships between the exocytic and endocytic apparatus at the synapse and provide a new conceptual framework to quantify synaptic architecture. 

Abstract We present 1–12 GHz Karl G. Jansky Very Large Array observations of nine off-nuclear persistent radio sources (PRSs) in nearby (z≲ 0.055) dwarf galaxies, along with high-resolution European VLBI Network observations for one of them at 1.7 GHz. We explore the plausibility that these PRSs are associated with fast radio burst (FRB) sources by examining their properties—physical sizes, host-normalized offsets, spectral energy distributions (SEDs), radio luminosities, and light curves—and compare them to those of the PRSs associated with FRB 20121102A and FRB 20190520B, two known active galactic nuclei (AGN), and one likely AGN in our sample with comparable data, as well as other radio transients exhibiting characteristics analogous to FRB-PRSs. We identify a single source in our sample, J1136+2643, as the most promising FRB-PRS, based on its compact physical size and host-normalized offset. We further identify two sources, J0019+1507 and J0909+5655, with physical sizes comparable to FRB-PRSs, but which exhibit large offsets and flat spectral indices potentially indicative of a background AGN origin. We test the viability of neutron star wind nebula and hypernebula models for J1136+2643 and find that the physical size, luminosity, and SED of J1136+2643 are broadly consistent with these models. Finally, we discuss the alternative interpretation that the radio sources are instead powered by accreting massive black holes, and we outline future prospects and follow-up observations for differentiating between these scenarios.