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A permutation in a finite symmetric group on a set of ordered elements has a descent at the i-th index if the permutation value at the i-th index is greater than the permutation value that follows. The descent set of a permutation is the set of all indices where the permutation has a descent. Each finite symmetric group can be partitioned by descent sets. In this paper we study the Hamming metric and the L-infinity metric on the sets of permutations that share the same descent set for all nonempty descent sets to determine the maximum possible value that these metrics can achieve when restricted to these subsets.  

The phenomenon of ionic liquid (IL) nanoconfinement within a copolymer/IL membrane reportedly enhances membrane selectivity, solubility, and transport in gas separations. Also, the copolymer/IL membrane morphology has been found to affect IL stability at high transmembrane pressures. In this work, a combined mesoscopic dynamics simulation and hybrid grand canonical Monte Carlo/molecular dynamics (GCMC-MD) simulations were carried out to investigate the morphologies, as well as CO2/CH4 gas diffusivities, solubilities, and selectivities of polystyrene-b-poly(ethylene oxide) (PS-b-PEO)/1-Ethyl-3-methylimidazolium thiocyanate ([EMIM][SCN]) and PS-b-PEO/1-Ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ([EMIM][Tf2N]) membranes. The latter simulations focused on nanoconfined ILs in the copolymer/IL phase boundaries at 2.5 and 5 nm confinement lengths. The investigated systems were four nanoconfined ILs, i.e., PS/[EMIM][SCN]/PEO (the IL forming a separate microphase, denoted IL-Micro), PS/[EMIM][Tf2N]/PEO, PS/[EMIM][SCN]-PEO/PS (the IL distributed in the PEO phase, denoted IL-PEO), and PS/[EMIM][Tf2N]-PEO/PS, and five control systems, i.e., PS/PEO/PS, bulk PS, bulk PEO, bulk [EMIM][SCN], and bulk [EMIM][Tf2N]. Based on the mesoscopic dynamics simulation results, the dominant membrane morphologies at IL loadings of <50 vol % were lamellar or cylindrical (favorable for both IL stability at high transmembrane pressures if the bedding planes are horizontal, i.e. at 90° to the nominal direction of the transmembrane pressure gradient) with the IL-PEO or IL-Micro phases. Also, there was an overall 50% match between the observed PS-b-PEO/[EMIM][SCN] and PS-b-PEO/[EMIM][Tf2N] membrane morphologies. Based on the MD simulation results, both CO2 and CH4 diffusivities were the smallest in the bulk PS (control) and highest in the PS/[EMIM][Tf2N]/PEO system (IL-Micro between the PS and PEO phases) at both confinement lengths. The CO2 diffusivities were, on average, larger when the confinement length increased to 5 nm. The GCMC-MD results indicated that the CO2 solubility in the IL-Micro phases was higher than in the corresponding bulk ILs at both confinement lengths, with the PS/[EMIM][Tf2N]/PEO system exhibiting the highest CO2 solubility, followed by the PS/[EMIM][SCN]/PEO system. Additionally, the permselectivities of the nanoconfined IL systems were, on average, 40–50% larger than those of the bulk systems, with the highest permselectivity observed for PS/[EMIM][Tf2N]/PEO at the confinement length of 5 nm. Overall, the IL nanoconfinement between the PS and PEO phases (IL-Micro) leads to significant improvements in the CO2/CH4 permselectivities, suggesting that strategies to create nanoconfined IL morphologies in the copolymer/IL membranes are very promising for optimizing the membrane gas separation performance. 

In mathematics, counter narratives can be used to fight the dominant narrative of who is good at mathematics and who can succeed in mathematics. Eight mathematicians were recruited to co-author a larger NSF project (RAMP). In part, they were asked to create author stories for an undergraduate audience. In this article, we use narrative analysis to present five polarities identified in the author stories. We present various quotations from the mathematicians’ author stories to highlight their experiences with home and school life, view of what mathematics is, experiences in growth in mathematics, with collaboration, and their feelings of community in mathematics. The telling of these experiences contributes towards rehumanizing mathematics and rewriting the narrative of who is good at and who can succeed in mathematics. 

An arithmetical structure on a finite, connected graph without loops is an assignment of positive integers to the vertices that satisfies certain conditions. Associated to each of these is a finite abelian group known as its critical group. We show how to determine the critical group of an arithmetical structure on a star graph or complete graph in terms of the entries of the arithmetical structure. We use this to investigate which finite abelian groups can occur as critical groups of arithmetical structures on these graphs. 

Abstract The Biorepository and Integrative Genomics (BIG) Initiative in Tennessee has developed a pioneering resource to address gaps in genomic research by linking genomic, phenotypic, and environmental data from a diverse Mid-South population, including underrepresented groups. We analyzed 13,152 exomes from BIG and found significant genetic diversity, with 50% of participants inferred to have non-European or several types of admixed ancestry. Ancestry within the BIG cohort is stratified, with distinct geographic and demographic patterns, as African ancestry is more common in urban areas, while European ancestry is more common in suburban regions. We observe ancestry-specific rates of novel genetic variants, which are enriched for functional or clinical relevance. Disease prevalence analysis linked ancestry and environmental factors, showing higher odds ratios for asthma and obesity in minority groups, particularly in the urban area. Finally, we observe discrepancies between self-reported race and genetic ancestry, with related individuals self-identifying in differing racial categories. These findings underscore the limitations of race as a biomedical variable. BIG has proven to be an effective model for community-centered precision medicine. We integrated genomics education, and fostered great trust among the contributing communities. Future goals include cohort expansion, and enhanced genomic analysis, to ensure equitable healthcare outcomes.