Abstract Arithmetic billiards show a nice interplay between arithmetics and geometry. The billiard table is rectangle with integer side lengths. A pointwise ball moves constant speed along segments making 45° angle the sides bounces on these. In classical setting, shooted from corner lands in corner. We allow to start at any point distances sides: either or trajectory periodic. length of path c...

Abstract Reinforcement learning has received much attention for controllers of deterministic systems. We consider a learner-verifier framework stochastic control systems and survey recent methods that formally guarantee conjunction reachability safety properties. Given property lower bound on the probability being satisfied, our jointly learns policy formal certificate to ensure satisfaction wi...

it is known that a stochastic dierential equation (sde) induces two probabilisticobjects, namely a diusion process and a stochastic ow. while the diusion process isdetermined by the innitesimal mean and variance given by the coecients of the sde,this is not the case for the stochastic ow induced by the sde. in order to characterize thestochastic ow uniquely the innitesimal covariance give...