Georgia Institute of Technology College of Sciences

We use a stochastic dynamic programming approach to address the following question: Can a homogenous resource extraction model (one without extraction costs, without new discoveries, and without technical progress) generate non-increasing resource prices? The traditional answer to that question contends that prices should exhibit an increasing trend as the exhaustible resource is being depleted over time (The Hotelling rule). In contrast, we will show that injecting concerns for temporal resolution of uncertainty in a resource extraction problem can generate a non-increasing trend in the resource price. Indeed, the expected rate of change of the price can become negative if the premium for temporal resolution of uncertainty is negative and outweighs both the positive discount rate and the short-run risk premium. Numerical examples are provided for illustration.

TBA

There are a number of ways to define the braid group. The traditional definition involves equivalence classes of braids, but it can also be defined in terms of mapping class groups, in terms of configuration spaces, or purely algebraically with an explicit presentation. My goal is to give an informal overview of this group and some of its subgroups, comparing and contrasting the various incarnations along the way.

We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - i.e. periodic and even in the space variable x - of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the quasi-linear nature of the gravity water waves equations and the fact that the linear frequencies grow just in a sublinear way at infinity. To overcome these problems, we first reduce the linearized operators obtained at each approximate quasi-periodic solution along the Nash-Moser iteration to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme that requires very weak Melnikov non-resonance conditions (which lose derivatives both in time and space), which we are able to verify for most values of the depth parameter using degenerate KAM theory arguments. This is a joint work with P. Baldi, M. Berti and R. Montalto.