Under the ringed basins

Basins of Attraction and Equilibrium Selection Under Different Learning Rules

A deterministic learning model applied to a game with multiple equilibria produces distinct basins of attraction for those equilibria. In symmetric two-by-two games, basins of attraction are invariant to a wide range of learning rules including best response dynamics, replicator dynamics, and fictitious play. In this paper, we construct a class of three-by-three symmetric games for which the ov...

Viability Kernels and Capture Basins of Sets Under Differential Inclusions

This paper provides a characterization of viability kernels and capture basins of a target viable in a constrained subset as a unique closed subset between the target and the constrained subset satisfying tangential conditions or, by duality, normal conditions. It is based on a method devised by Hélène Frankowska for characterizing the value function of an optimal control problem as generalized...

The allure of the ringed planets

One of the most enduring symbols of space exploration is a planet surrounded by a ring. This symbol inspires a celestial context: nothing on Earth is like it. My own fascination with space began when first I saw Saturn and its rings through a small telescope. The study of planets and rings has become my lifelong vocation. What a wonderful surprise that the ringed planets are just as beautiful a...

Generalized Ringed Spaces and Schemes

Modules over a Ringed Space

In these notes we collect some useful facts about sheaves of modules on a ringed space that are either left as exercises in [Har77] or omitted completely. Most of these observations can be found in [Gro60], which is of course always the best reference. Nonetheless it may be useful for the reader to have an english reference when these points arise in our other notes on algebraic geometry. These...