A new resummation scheme in scalar field theories

A new resummation scheme in scalar field theories is proposed by combining parquet resummation techniques and flow equations, which is characterized by a hierarchy structure of the Bethe--Salpeter (BS) equations. The new resummation scheme greatly improves on the approximations for the BS kernel. Resummation of the BS kernel in the $t$ and $u$ channels to infinite order is equivalent to truncate the effective action to infinite order. Our approximation approaches ensure that the theory can be renormalized, which is very important for numerical calculations. Two-point function can also be obtained from the four-point one through flow evolution equations resulting from the functional renormalization group. BS equations of different hierarchies and the flow evolution equation for the propagator constitute a closed self-consistent system, which can be solved completely.Comment: 12 pages, 4 figures. v2: title changed, one figure added, final version in Eur. Phys. J.

Probably Approximately Correct MDP Learning and Control With Temporal Logic Constraints

We consider synthesis of control policies that maximize the probability of satisfying given temporal logic specifications in unknown, stochastic environments. We model the interaction between the system and its environment as a Markov decision process (MDP) with initially unknown transition probabilities. The solution we develop builds on the so-called model-based probably approximately correct Markov decision process (PAC-MDP) methodology. The algorithm attains an $\varepsilon$-approximately optimal policy with probability $1-\delta$ using samples (i.e. observations), time and space that grow polynomially with the size of the MDP, the size of the automaton expressing the temporal logic specification, $\frac{1}{\varepsilon}$, $\frac{1}{\delta}$ and a finite time horizon. In this approach, the system maintains a model of the initially unknown MDP, and constructs a product MDP based on its learned model and the specification automaton that expresses the temporal logic constraints. During execution, the policy is iteratively updated using observation of the transitions taken by the system. The iteration terminates in finitely many steps. With high probability, the resulting policy is such that, for any state, the difference between the probability of satisfying the specification under this policy and the optimal one is within a predefined bound.Comment: 9 pages, 5 figures, Accepted by 2014 Robotics: Science and Systems (RSS