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In this paper, we construct two spanning sets for the affine algebraic curvature tensors. We then prove that every 2-dimensional affine algebraic curvature tensor can be represented by a single element from either of the two spanning sets. This paper provides a means to study affine algebraic curvature tensors in a geometric and algebraic manner similar to previous studies of canonical algebraic curvature tensors. 

Abstract The availability of dissolved inorganic and organic nutrients and their transformations along the fresh to marine continuum are being modified by various natural and anthropogenic activities and climate-related changes. Subtropical central and eastern Florida Bay, located at the southern end of the Florida peninsula, is classically considered to have inorganic nutrient conditions that are in higher-than-Redfield ratio proportions, and high levels of organic and chemically-reduced forms of nitrogen. However, salinity, pH and nutrients, both organic and inorganic, change with changes in freshwater flows to the bay. Here, using a time series of water quality and physico-chemical conditions from 2009 to 2019, the impacts of distinct changes in managed flow, drought, El Niño-related increases in precipitation, and intensive storms and hurricanes are explored with respect to changes in water quality and resulting ecosystem effects, with a focus on understanding why picocyanobacterial blooms formed when they did. Drought produced hyper-salinity conditions that were associated with a seagrass die-off. Years later, increases in precipitation resulting from intensive storms and a hurricane were associated with high loads of organic nutrients, and declines in pH, likely due to high organic acid input and decaying organic matter, collectively leading to physiologically favorable conditions for growth of the picocyanobacterium, Synechococcus spp. These conditions, including very high concentrations of NH 4 + , were likely inhibiting for seagrass recovery and for growth of competing phytoplankton or their grazers. Given projected future climate conditions, and anticipated cycles of drought and intensive storms, the likelihood of future seagrass die-offs and picocyanobacterial blooms is high. 

It has been presumed that rheumatoid arthritis (RA) joint pain is related to inflammation in the synovium; however, recent studies reveal that pain scores in patients do not correlate with synovial inflammation. We developed a machine-learning approach (graph-based gene expression module identification or GbGMI) to identify an 815-gene expression module associated with pain in synovial biopsy samples from patients with established RA who had limited synovial inflammation at arthroplasty. We then validated this finding in an independent cohort of synovial biopsy samples from patients who had early untreated RA with little inflammation. Single-cell RNA sequencing analyses indicated that most of these 815 genes were most robustly expressed by lining layer synovial fibroblasts. Receptor-ligand interaction analysis predicted cross-talk between human lining layer fibroblasts and human dorsal root ganglion neurons expressing calcitonin gene–related peptide (CGRP⁺). Both RA synovial fibroblast culture supernatant and netrin-4, which is abundantly expressed by lining fibroblasts and was within the GbGMI-identified pain-associated gene module, increased the branching of pain-sensitive murine CGRP⁺dorsal root ganglion neurons in vitro. Imaging of solvent-cleared synovial tissue with little inflammation from humans with RA revealed CGRP⁺pain-sensing neurons encasing blood vessels growing into synovial hypertrophic papilla. Together, these findings support a model whereby synovial lining fibroblasts express genes associated with pain that enhance the growth of pain-sensing neurons into regions of synovial hypertrophy in RA.