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We define a new gauge independent quasi-local mass and energy, and show its relation to the Brown–York Hamilton–Jacobi analysis. A quasi-local proof of the positivity, based on spacetime harmonic functions, is given for admissible closed spacelike 2-surfaces which enclose an initial data set satisfying the dominant energy condition. Like the Wang-Yau mass, the new definition relies on isometric embeddings into Minkowski space, although our notion of admissibility is different from that of Wang and Yau. Rigidity is also established, in that vanishing energy implies that the 2-surface arises from an embedding into Minkowski space, and conversely the mass vanishes for any such surface. Furthermore, we show convergence to the ADM mass at spatial infinity, and provide the equation associated with optimal isometric embedding. 

Interactions among the underlying agents of a complex system are not only limited to dyads but can also occur in larger groups. Currently, no generic model has been developed to capture high-order interactions (HOI), which, along with pairwise interactions, portray a detailed landscape of complex systems. Here, we integrate evolutionary game theory and behavioral ecology into a unified statistical mechanics framework, allowing all agents (modeled as nodes) and their bidirectional, signed, and weighted interactions at various orders (modeled as links or hyperlinks) to be coded into hypernetworks. Such hypernetworks can distinguish between how pairwise interactions modulate a third agent (active HOI) and how the altered state of each agent in turn governs interactions between other agents (passive HOI). The simultaneous occurrence of active and passive HOI can drive complex systems to evolve at multiple time and space scales. We apply the model to reconstruct a hypernetwork of hexa-species microbial communities, and by dissecting the topological architecture of the hypernetwork using GLMY homology theory, we find distinct roles of pairwise interactions and HOI in shaping community behavior and dynamics. The statistical relevance of the hypernetwork model is validated using a series of in vitro mono-, co-, and tricultural experiments based on three bacterial species. 

Abstract LetXbe a compact, Kähler, Calabi‐Yau threefold and suppose , for , is a conifold transition obtained by contracting finitely many disjoint curves inXand then smoothing the resulting ordinary double point singularities. We show that, for sufficiently small, the tangent bundle admits a Hermitian‐Yang‐Mills metric with respect to the conformally balanced metrics constructed by Fu‐Li‐Yau. Furthermore, we describe the behavior of near the vanishing cycles of as . 

Abstract We introduce a notion of ‘cross-section continuity’ as a criterion for the viability of definitions of angular momentum, J , at null infinity: If a sequence of cross-sections, , of null infinity converges uniformly to a cross-section , then the angular momentum, J n , on should converge to the angular momentum, J , on . The Dray–Streubel (DS) definition of angular momentum automatically satisfies this criterion by virtue of the existence of a well defined flux associated with this definition. However, we show that the one-parameter modification of the DS definition proposed by Compere and Nichols—which encompasses numerous other alternative definitions—does not satisfy cross-section continuity. On the other hand, we prove that the Chen–Wang–Yau definition does satisfy the cross-section continuity criterion.