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Abstract PremiseStudies into the evolution and development of leaf shape have connected variation in plant form, function, and fitness. For species with consistent leaf margin features, patterns in leaf architecture are related to both biotic and abiotic factors. However, for species with inconsistent leaf shapes, quantifying variation in leaf shape and the effects of environmental factors on leaf shape has proven challenging. MethodsTo investigate leaf shape variation in a species with inconsistently shaped leaves, we used geometric morphometric modeling and deterministic techniques to analyze approximately 500 digitized specimens ofCapsella bursa‐pastoriscollected throughout the continental United States over 100 years. We generated a morphospace of the leaf shapes and modeled leaf shape as a function of environment and time. ResultsLeaf shape variation ofC. bursa‐pastoriswas strongly associated with temperature over its growing season, with lobing decreasing as temperature increased. While we expected to see changes in variation over time, our results show that the level of leaf shape variation was consistent over the 100 years. ConclusionsOur findings showed that species with inconsistent leaf shape variation can be quantified using geometric morphometric modeling techniques and that temperature is the main environmental factor influencing leaf shape variation.
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Angle ranks of abelian varieties
Abstract Using the formalism of Newton hyperplane arrangements, we resolve the open questions regarding angle rank left over from work of the first two authors with Roe and Vincent. As a consequence we end up generalizing theorems of Lenstra–Zarhin and Tankeev proving several new cases of the Tate conjecture for abelian varieties over finite fields. We also obtain an effective version of a recent theorem of Zarhin bounding the heights of coefficients in multiplicative relations among Frobenius eigenvalues.
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- Award ID(s):
- 2053473
- PAR ID:
- 10418453
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Mathematische Annalen
- Volume:
- 389
- Issue:
- 1
- ISSN:
- 0025-5831
- Format(s):
- Medium: X Size: p. 169-185
- Size(s):
- p. 169-185
- Sponsoring Org:
- National Science Foundation
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