Terrestrial ecosystem productivity is widely accepted to be nutrient limited(1). Although nitrogen (N) is deemed a key determinant of aboveground net primary production (ANPP)(2,3), the prevalence of co-limitation by N and phosphorus (P) is increasingly recognized(4-8). However, the extent to which terrestrial productivity is co-limited by nutrients other than N and P has remained unclear. Here, we report results from a standardized factorial nutrient addition experiment, in which we added N, P and potassium (K) combined with a selection of micronutrients (K+μ), alone or in concert, to 42 grassland sites spanning five continents, and monitored ANPP. Nutrient availability limited productivity at 31 of the 42 grassland sites. And pairwise combinations of N, P, and K+μ co-limited ANPP at 29 of the sites. Nitrogen limitation peaked in cool, high latitude sites. Our findings highlight the importance of less studied nutrients, such as K and micronutrients, for grassland productivity, and point to significant variations in the type and degree of nutrient limitation. We suggest that multiple-nutrient constraints must be considered when assessing the ecosystem-scale consequences of nutrient enrichment.

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Recommending purely cold-start items is a long-standing and fundamental challenge in the recommender systems. Without any historical interaction on cold-start items, the collaborative filtering (CF) scheme fails to leverage collaborative signals to infer user preference on these items. To solve this problem, extensive studies have been conducted to incorporate side information of items (e.g., content features) into the CF scheme. Specifically, they employ modern neural network techniques (e.g., dropout, consistency constraint) to discover and exploit the coalition effect of content features and collaborative representations. However, we argue that these works less explore the mutual dependencies between content features and collaborative representations and lack sufficient theoretical supports, thus resulting in unsatisfactory performance on cold-start recommendation.In this work, we reformulate the cold-start item representation learning from an information-theoretic standpoint. It aims to maximize the mutual dependencies between item content and collaborative signals. Specifically, the representation learning is theoretically lower-bounded by the integration of two terms: mutual information between collaborative embeddings of users and items, and mutual information between collaborative embeddings and feature representations of items. To model such a learning process, we devise a new objective function founded upon contrastive learning and develop a simple yet efficient Contrastive Learning-based Cold-start Recommendation framework (CLCRec). In particular, CLCRec consists of three components: contrastive pair organization, contrastive embedding, and contrastive optimization modules.

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