the notion of a probabilistic metric  space  corresponds to thesituations when we do not know exactly the  distance.  probabilistic metric space was  introduced by karl menger. alsina, schweizer and sklar gave a general definition of  probabilistic normed space based on the definition of menger [1]. in this note we study the pn spaces which are  topological vector spaces and the open mapping an...

In this paper, the connection between Menger probabilistic norms and H"{o}hle probabilistic norms is discussed. In addition, the correspondence between probabilistic norms and Wu-Fang fuzzy (semi-) norms is established. It is shown that a probabilistic norm (with triangular norm $min$) can generate a Wu-Fang fuzzy semi-norm and conversely, a Wu-Fang fuzzy norm can generate a probabilistic norm.

Abstract We present an adaptive grid refinement algorithm to solve probabilistic optimization problems with infinitely many random constraints. Using a bilevel approach, we iteratively aggregate inequalities that provide most information not in geometric but sense. This conceptual idea, for which convergence proof is provided, then adapted implementable algorithm. The efficiency of our approach...

We propose to leverage epistemic uncertainty about constraint satisfaction of a reinforcement learner in safety critical domains. introduce framework for specification requirements learners constrained settings, including confidence results. show that an agent's provides useful signal balancing optimization and the learning process.

We consider probabilistic allocation of objects under ordinal preferences and constraints on allocations. devise an mechanism, called the vigilant eating rule (VER), that applies to nearly arbitrary constraints. It is constrained ordinally efficient, can be computed efficiently for a large class constraints, treats agents equally if they have same are subject When set feasible allocations conve...