Contact Structures on Plumbed 3-Manifolds

In this paper, we show that the Ozsv\'ath-Szab\'o contact invariant $c^+(\xi)\in HF^+(-Y)$ of a contact 3-manifold $(Y,\xi)$ can be calculated combinatorially if $Y$ is the boundary of a certain type of plumbing $X$, and $\xi$ is induced by a Stein structure on $X$. Our technique uses an algorithm of Ozsv\'ath and Szab\'o to determine the Heegaard-Floer homology of such 3-manifolds. We discuss two important applications of this technique in contact topology. First, we show that it simplifies the calculation of the Ozsv\'ath-Stipsicz-Szab\'o obstruction to admitting a planar open book. Then we define a numerical invariant of contact manifolds that respects a partial ordering induced by Stein cobordisms. We do a sample calculation showing that the invariant can get infinitely many distinct values.Comment: Added some examples and comment

Optimal pattern of technology adoption under embodiment with a finite planning horizon : A multi-stage optimal control approach

By deriving the necessary conditions for a multi-stage discounted optimal control problem where the endogenous switching instants between regimes appear as an argument of the objective function and the state equation, we analyze the optimal pattern of technology adoption under embodiment with a finite planning horizon. The economy is characterized by the existence of an exogenously growing technology frontier and technology specific learning by doing. We obtain time varying durations for the adopted technologies to be in use due to finite planning horizon. We analyze numerically the effects of planing horizon, speed of learning, growth rate of technology and impatience rate on the optimal pattern.Multi-stage optimal control, technology adoption, learning by doing, embodiment