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Abstract We develop and study a generalization of commutative rings calledbands, along with the corresponding geometric theory ofband schemes. Bands generalize both hyperrings, in the sense of Krasner, and partial fields in the sense of Semple and Whittle. They form a ring‐like counterpart to the field‐like category ofidyllsintroduced by the first and third authors in the previous work. The first part of the paper is dedicated to establishing fundamental properties of bands analogous to basic facts in commutative algebra. In particular, we introduce various kinds of ideals in a band and explore their properties, and we study localization, quotients, limits, and colimits. The second part of the paper studies band schemes. After giving the definition, we present some examples of band schemes, along with basic properties of band schemes and morphisms thereof, and we describe functors into some other scheme theories. In the third part, we discuss some “visualizations” of band schemes, which are different topological spaces that one can functorially associate to a band scheme . 

Thefoundationof a matroid is a canonical algebraic invariant which classifies, in a certain precise sense, all representations of the matroid up to rescaling equivalence. Foundations of matroids arepastures, a simultaneous generalization of partial fields and hyperfields which are special cases of both tracts (as defined by the first author and Bowler) and ordered blue fields (as defined by the second author). Using deep results due to Tutte, Dress–Wenzel, and Gelfand–Rybnikov–Stone, we give a presentation for the foundation of a matroid in terms of generators and relations. The generators are certain “cross-ratios” generalizing the cross-ratio of four points on a projective line, and the relations encode dependencies between cross-ratios in certain low-rank configurations arising in projective geometry. Although the presentation of the foundation is valid for all matroids, it is simplest to apply in the case of matroidswithout large uniform minors. i.e., matroids having no minor corresponding to five points on a line or its dual configuration. For such matroids, we obtain a complete classification of all possible foundations. We then give a number of applications of this classification theorem, for example: We prove the following strengthening of a 1997 theorem of Lee and Scobee: every orientation of a matroid without large uniform minors comes from a dyadic representation, which is unique up to rescaling. For a matroid $M$without large uniform minors, we establish the following strengthening of a 2017 theorem of Ardila–Rincón–Williams: if $M$is positively oriented then $M$is representable over every field with at least 3 elements. Two matroids are said to belong to the samerepresentation classif they are representable over precisely the same pastures. We prove that there are precisely 12 possibilities for the representation class of a matroid without large uniform minors, exactly three of which are not representable over any field. 

Abstract We give a new proof, along with some generalizations, of a folklore theorem (attributed to Laurent Lafforgue) that a rigid matroid (i.e., a matroid with indecomposable basis polytope) has only finitely many projective equivalence classes of representations over any given field. 

Abstract ContextLand use history of urban forests impacts present-day soil structure, vegetation, and ecosystem function, yet is rarely documented in a way accessible to planners and land managers. ObjectivesTo (1) summarize historical land cover of present-day forest patches in Baltimore, MD, USA across land ownership categories and (2) determine whether social-ecological characteristics vary by historical land cover trajectory. MethodsUsing land cover classification derived from 1927 and 1953 aerial imagery, we summarized present-day forest cover by three land cover sequence classes: (1) Persistent forest that has remained forested since 1927, (2) Successional forest previously cleared for non-forest vegetation (including agriculture) that has since reforested, or (3) Converted forest that has regrown on previously developed areas. We then assessed present-day ownership and average canopy height of forest patches by land cover sequence class. ResultsMore than half of Baltimore City’s forest has persisted since at least 1927, 72% since 1953. About 30% has succeeded from non-forest vegetation during the past century, while 15% has reverted from previous development. A large proportion of forest converted from previous development is currently privately owned, whereas persistent and successional forest are more likely municipally-owned. Successional forest occurred on larger average parcels with the fewest number of distinct property owners per patch. Average tree canopy height was significantly greater in patches of persistent forest (mean = 18.1 m) compared to canopy height in successional and converted forest patches (16.6 m and 16.9 m, respectively). ConclusionsHistorical context is often absent from urban landscape ecology but provides information that can inform management approaches and conservation priorities with limited resources for sustaining urban natural resources. Using historical landscape analysis, urban forest patches could be further prioritized for protection by their age class and associated ecosystem characteristics. 

Plant traits are often measured in the field or laboratory to characterize stress responses. However, direct measurements are not always cost effective for broader sampling efforts, whereas indirect approaches such as reflectance spectroscopy could offer efficient and scalable alternatives. Here, we used field spectroscopy to assess whether (1) existing vegetation indices could predict leaf trait responses to heat stress, or if (2) partial least squares regression (PLSR) spectral models could quantify these trait responses. On several warm, sunny days, we measured leaf trait responses indicative of photosynthetic mechanisms, plant water status, and morphology, including electron transport rate (ETR), photochemical quenching (qP), leaf water potential (Ψleaf), and specific leaf area (SLA) in 51 urban trees from nine species. Concurrent measures of hyperspectral leaf reflectance from the same individuals were used to calculate vegetation indices for correlation with trait responses. We found that vegetation indices predicted only SLA robustly (R2 = 0.55), while PLSR predicted all leaf trait responses of interest with modest success (R2 = 0.36 to 0.58). Using spectral band subsets corresponding to commercially available drone-mounted hyperspectral cameras, as well as those selected for use in common multispectral satellite missions, we were able to estimate ETR, qP, and SLA with reasonable accuracy, highlighting the potential for large-scale prediction of these parameters. Overall, reflectance spectroscopy and PLSR can identify wavelengths and wavelength ranges that are important for remote sensing-based modeling of important functional trait responses of trees to heat stress over broad ranges.