Abstract. Choo-Siow (2006) proposed a model for the marriage market which allows for random identically distributed noise in the preferences of each of the participants. The randomness is McFadden-type, which permits an explicit resolution of the equilibrium preference probabilities.The purpose of this note is to prove uniqueness of the resulting equilibrium marriage distribution, and find a representation of it in closed form. This allows us to derive smooth dependence of this distribution on exogenous preference and population parameters, and establish sign, symmetry, and size of the various substitution effects, facilitating comparative statics. For example, we show that an increase in the population of men of any given type in this model leads to an increase in single men of each type, and a decrease in single women of each type. We show that an increase in the number of men of a given type increases the equilibrium transfer paid by such men to their spouses, and also increases the percentage of men of that type who choose to remain unmarried. While the above trends may not seem surprising, the verification of such properties helps to substantiate the validity of the model. Moreover, we make unexpected predictions which could be tested: namely, the percentage change of type i unmarrieds with respect to fluctuations in the total number of type j men or women turns out to form a symmetric positive-definite matrix rij = rji in this model, and thus to satisfy bounds such as |rij | ≤ (riirjj ) 1/2 .Along the way, we give a new proof for the existence of an equilibrium, based on a strictly convex variational principle and a simple estimate, rather than a fixed point theorem. Fixed point approaches to the existence part of our result have been explored by others [6] [8] [12], but are much more complicated and yield neither uniqueness, nor comparative statics, nor an explicit representation of the solution.JEL Classification: J12, C62, C78, C81, D03, Z13 KEYWORDS: Choo-Siow, marriage market, matching, random, unique equilibrium, comparative statics, convex analysis.Date: March 27, 2018. The authors are grateful to Aloysius Siow for attracting their attention to this question, and for many fruitful discussions. This project developed in part from the Master's research of CD; however, the present manuscript is based a new approach which greatly extends (and largely subsumes) the results of his thesis [9], and of the 2009 preprint by three of us entitled When do systematic gains uniquely determine the number of marriages between different types in the Choo-Siow matching model? Sufficient conditions for a unique equilibrium. CD, RJM and BKS are pleased to acknowledge the support of Natural Sciences and Engineering

We study an energy functional that arises as a Γ-limit of the Gross-Pitaevskii (GP) energy. This last functional is often used to model rotating Bose-Einstein condensates, and the functional we study represents the contribution to the GP energy of vortices, or whirlpools, in the condensate. For our energy, we give a rough description of its (local) minimizers using ODE techniques along with an isoperimetric inequality.

The degenerate parabolic equation u t + ∂ x [u 3 (u xxx + u x − sin x)] = 0 models the evolution of a thin liquid film on a stationary horizontal cylinder. It is shown here that for each mass there is a unique steady state, given by a droplet hanging from the bottom of the cylinder that meets the dry region with zero contact angle. The droplet minimizes the associated energy functional and attracts all strong solutions that satisfy certain energy and entropy inequalities, including all positive solutions. The distance of solutions from the steady state cannot decay faster than a power law.

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