We examine a dynamic, two‐sided, one‐to‐one matching market where agents on both sides interact over a period of time. We define and identify sufficient conditions for the existence of a dynamically stable matching, which may require revisions to initial assignments. A generalization of the deferred acceptance algorithm can identify dynamically stable outcomes in a large class of economies, including cases with intertemporal preference complementarities. We relate our analysis to market unraveling and to common market design applications, including the medical residency match.

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Liquid-liquid phase separation (LLPS) and subsequent liquid-to-solid transition is implicated in membraneless organelles formation as well as disease associated protein aggregation. However, how liquid-to-solid transition is initiated inside a liquid droplet remains unclear. Here, using studies at single droplet resolution, we show that liquid-to-solid transition of α-synuclein (α-Syn) liquid droplets is associated with significant changes in the local microenvironment as well as secondary structure of the protein, which is prominently observed at the center of the liquid droplets. With the ageing of liquid droplets, the structured core at the center gradually expands and propagates over entire droplets. Further, during droplet fusion, smaller, homogeneous droplets progressively dissolve and supply proteins to the larger, heterogeneous droplets containing solid-like core at their center. The present study will significantly help to under-stand the physical mechanism of LLPS and liquid-to-solid transition in biological compartmentalization as well as in protein aggregation associated with human neurodegenerative disorders.

We prove that the unilateral substitutability property introduced in Hatfield and Kojima [2010] implies the substitutable completability property from Hatfield and Kominers [2014]. This paper provides a novel linkage between these two sufficient conditions for the existence of a stable matching in many-to-one matching markets with contracts. A substitutable completion of a preference is a substitutable preference created by adding some sets of contracts to the original preference order. We provide an algorithm which when operated on the unilaterally substitutable preferences produces such a substitutable completion. Thus it provides a constructive proof of the connection between the two properties. * I am deeply indebted to Scott Kominers and Eric Maskin for extremely helpful discussions and detailed comments on this paper. I would also like to thank