The Uniqueness of a Mean Field Equilibrium

We present proofs of two classical theorems. The first one, due to Darboux and Sauer, states that infinitesimal rigidity is a projective invariant; the other one establishes relations (infinitesimal Pogorelov maps) between the infinitesimal motions of a Euclidean framework and of its hyperbolic and spherical images. The arguments use the static formulation of infinitesimal rigidity. The duality...

This paper analyzes monotone comparative statics predictions in several classes of stochastic optimization problems. The main results characterize necessary and sufficient conditions for comparative statics predictions to hold based on properties of primitive functions, that is, utility functions and probability distributions. The results apply when the primitives satisfy one of the following t...

I prove that, in models with complementarities, some non-monotone comparative statics must select unstable equilibria; and, under additional regularity conditions, that monotone comparative statics selects stable equilibria. JEL Classification. C61, C62, C72, C73. Discussions with my advisers, Ilya Segal and Chris Shannon, were very important for this work. I am very grateful for their help. I ...

We analyze comparative risk aversion in a new way, through a comparative statics problem in which, for a cost, agents can shift from an initial probability distribution toward a preferred distribution. The Ross characterization arises when the original distribution is riskier than the preferred distribution and the cost is monetary, and theArrow–Pratt characterization ariseswhen the original di...

Multi-component recording contains full wavefield information that can be used in polarization filtering to attenuate surface-wave noise. The polarization filter used here is a station-by-station approach that combines statics, signal-to-noise ratio, and eigenimages to construct a surfacewave noise model and then applies an adaptive subtraction to remove the noise. Statics and signal-to-noise r...

The classic theory of differential geometry concerns itself with smooth curves and surfaces. In practice, however, our experiments can only measure a finite amount of data, and our simulations can only resolve a finite amount of detail. Discrete differential geometry (DDG) studies discrete counterparts of classical differential geometry that are applicable in this discrete setting, and converge...