Mechanisms, counterfactuals and laws

In this chapter we examine the relation between mechanisms and laws/counterfactuals by revisiting the main notions of mechanism found in the literature. We distinguish between two different conceptions of ‘mechanism’: mechanisms-of underlie or constitute a causal process; mechanisms-for are complex systems that function so as to produce a certain behavior. According to some mechanists, a mechanism fulfills both of these roles simultaneously. The main argument of the chapter is that there is an asymmetrical dependence between both kinds of mechanisms and laws/counterfactuals: while some laws and counterfactuals must be taken as primitive (non-mechanistic) facts of the world, all mechanisms depend on laws/counterfactuals

Rule Value Reinforcement Learning for Cognitive Agents

RVRL (Rule Value Reinforcement Learning) is a new algorithm which extends an existing learning framework that models the environment of a situated agent using a probabilistic rule representation. The algorithm attaches values to learned rules by adapting reinforcement learning. Structure captured by the rules is used to form a policy. The resulting rule values represent the utility of taking an action if the rule`s conditions are present in the agent`s current percept. Advantages of the new framework are demonstrated, through examples in a predator-prey environment

On the best constant of Hardy-Sobolev Inequalities

We obtain the sharp constant for the Hardy-Sobolev inequality involving the distance to the origin. This inequality is equivalent to a limiting Caffarelli-Kohn-Nirenberg inequality. In three dimensions, in certain cases the sharp constant coincides with the best Sobolev constant

Improving L2L^2 estimates to Harnack inequalities

We consider operators of the form ${\mathcal L}=-L-V$, where $L$ is an elliptic operator and $V$ is a singular potential, defined on a smooth bounded domain $\Omega\subset \R^n$ with Dirichlet boundary conditions. We allow the boundary of $\Omega$ to be made of various pieces of different codimension. We assume that ${\mathcal L}$ has a generalized first eigenfunction of which we know two sided estimates. Under these assumptions we prove optimal Sobolev inequalities for the operator ${\mathcal L}$, we show that it generates an intrinsic ultracontractive semigroup and finally we derive a parabolic Harnack inequality up to the boundary as well as sharp heat kernel estimates