We will show that group exactness is a von Neumann equivalence invariant. This result generalizes the previously known fact stating stable under measure and W*-equivalence.

We prove that a Poisson structure on projective toric variety which is invariant by the torus action and whose symplectic leaves are orbits not exact. This deduced from geometric criterion for non-exactness of structures with finite number leaves.

The optimal hypothesis tests for the binomial distribution and some other discrete distributions are uniformly most powerful (UMP) one-tailed and UMP unbiased (UMPU) two-tailed randomized tests. Conventional confidence intervals are not dual to randomized tests and perform badly on discrete data at small and moderate sample sizes. We introduce a new confidence interval notion, called fuzzy conf...