We consider three theorems in ergodic theory concerning a fixed aperiodic measure preserving transformation σ of a Lebesgue probability space (X,A, μ) and show that these theorems are all equivalent. Two of these results concern the existence of a partition of the space X with special properties. The third theorem asserts that the conjugates of σ are dense in the uniform topology on the space o...

A high-order accurate quadrature rule for the discretization of boundary integral equations (BIEs) on closed smooth contours in plane is introduced. This can be viewed as a hybrid spectral Kress (Math. Comput. Model. 15(3-5), 229–243 1991) and locally corrected trapezoidal Kapur Rokhlin (SIAM J. Numer. Anal. 34(4), 1331–1356, 1997). The new technique combines strengths both methods, attains con...

Using new spaces of tracial non-commutative smooth functions, we formulate a free probabilistic analog the Wasserstein manifold on $\mathbb{R}^d$ (the formal Riemannian probability densities $\mathbb{R}^d$), and use it to s