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The singular set B={x|F′(x)B = \\{ x|F’(x) is not surjective} of a nonlinear Fredholm operator F of positive index (between Banach spaces X1{X_1} and X2{X_2}) is investigated. Under the assumption that the mapping is proper and has a locally Lipschitzian Fréchet derivative F′(x)F’(x), it is shown that the singular set B is nonempty. Furthermore, when the Banach spaces are infinite dimensional, B cannot be the countable union of compact sets nor can F−1(F(B)){F^{ - 1}}(F(B)) contain isolated points.

Mitogen-activated protein kinase kinase kinases (MAPKKKs), which function at the top level of mitogen-activated protein kinase cascades, are clustered into three groups. However, no Group C Raf-like MAPKKKs have yet been functionally identified. We report here the characterization of a rice (Oryza sativa) mutant, increased leaf angle1 (ila1), resulting from a T-DNA insertion in a Group C MAPKKK gene. The increased leaf angle in ila1 is caused by abnormal vascular bundle formation and cell wall composition in the leaf lamina joint, as distinct from the mechanism observed in brassinosteroid-related mutants. Phosphorylation assays revealed that ILA1 is a functional kinase with Ser/Thr kinase activity. ILA1 is predominantly resident in the nucleus and expressed in the vascular bundles of leaf lamina joints. Yeast two-hybrid screening identified six closely related ILA1 interacting proteins (IIPs) of unknown function. Using representative IIPs, the interaction of ILA1 and IIPs was confirmed in vivo. IIPs were localized in the nucleus and showed transactivation activity. Furthermore, ILA1 could phosphorylate IIP4, indicating that IIPs may be the downstream substrates of ILA1. Microarray analyses of leaf lamina joints provided additional evidence for alterations in mechanical strength in ila1. ILA1 is thus a key factor regulating mechanical tissue formation at the leaf lamina joint.

Strawberry growers in Florida suffer from a lack of efficient and accurate yield forecasts for strawberries, which would allow them to allocate optimal labor and equipment, as well as other resources for harvesting, transportation, and marketing. Accurate estimation of the number of strawberry flowers and their distribution in a strawberry field is, therefore, imperative for predicting the coming strawberry yield. Usually, the number of flowers and their distribution are estimated manually, which is time-consuming, labor-intensive, and subjective. In this paper, we develop an automatic strawberry flower detection system for yield prediction with minimal labor and time costs. The system used a small unmanned aerial vehicle (UAV) (DJI Technology Co., Ltd., Shenzhen, China) equipped with an RGB (red, green, blue) camera to capture near-ground images of two varieties (Sensation and Radiance) at two different heights (2 m and 3 m) and built orthoimages of a 402 m2 strawberry field. The orthoimages were automatically processed using the Pix4D software and split into sequential pieces for deep learning detection. A faster region-based convolutional neural network (R-CNN), a state-of-the-art deep neural network model, was chosen for the detection and counting of the number of flowers, mature strawberries, and immature strawberries. The mean average precision (mAP) was 0.83 for all detected objects at 2 m heights and 0.72 for all detected objects at 3 m heights. We adopted this model to count strawberry flowers in November and December from 2 m aerial images and compared the results with a manual count. The average deep learning counting accuracy was 84.1% with average occlusion of 13.5%. Using this system could provide accurate counts of strawberry flowers, which can be used to forecast future yields and build distribution maps to help farmers observe the growth cycle of strawberry fields.

We present a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to construct maps to real and complex projective spaces, given appropriate persistent cohomology classes. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Next, the transition functions are used to produce explicit classifying maps for the induced bundles. A framework for dimensionality reduction in projective space (Principal Projective Components) is also developed, aimed at decreasing the target dimension of the original map. Several examples are provided as well as theorems addressing choices in the construction.

Quantifying the trade-offs and synergies among ecosystem services (ESs) and exploring their spatial scale effects are essential to guide sustainable and diversified ecosystem management. This study systematically analyzed the trade-offs and synergies between seven ESs, including water purification (WP), carbon storage (CS), habitat quality (HQ), net primary productivity (NPP), soil conservation (SC), water conservation (WC), and water yield (WY) using self-organizing feature map methods. Further, we used geographically weighted regression to quantify the effect of spatial scales (county and sub-watershed) on the trade-offs and synergies (bundles) of ESs and their spatial distribution in the Huaihe river basin from 2000 to 2020. The results indicated that (1) from 2000 to 2020, WP, NPP, and WC showed an upward trend, with the most significant increase in WC (an average increase of 15.03 mm). CS and HQ showed a downward trend. (2) The relationship between CS, HQ, NPP, SC, and WC was synergistic at both County and sub-watershed scales, and there was a substantial trade-off between WP and WY. from 2000 to 2020, the average correlation coefficients at the county scale and the sub-watershed scale were − 0.546 and − 0.434 respectively ( p  < 0.001). the synergy between CS, HQ, NPP, SC, and WC mainly occurred in mountainous and hilly areas, while the trade-off between WP and other ESs mainly appeared in the central plains. The relationship between WP and other ESs developed in the trade-off direction, and the other ES optimized in the collaborative direction. In terms of spatial distribution, the synergy area between most ESs is larger than the trade-off area, and the synergy area between HQ-NPP and WC-WY at the county scale is significantly larger than that at the sub-watershed scale. The average synergy area of each ES pair at the county scale is 20.48% larger than that at the sub-watershed scale. (3) Different ecosystem combinations provided six and eight ES bundles that differed in ES composition and quantity at the county and sub-watershed scales. The key synergetic bundle in the Southern Tongbai Dabie mountain is shrinking, which is more evident at the sub-watershed scale. the discovered spatial patterns, dynamics, and scale effects of ecosystem services provide helpful baseline information for exploring sustainable ecosystem management and cross-scale planning strategies in the Huaihe river basin.