In this article, we propose a general framework for distribution-free nonparametric testing in multi-dimensions, based on notion of multivariate ranks defined using the theory measure transportation. Unlike other existing proposals literature, these share number useful properties with usual one-dimensional ranks; most importantly, are distribution-free. This crucial observation allows us to des...

Symmetric trace-free tensors are used in many areas of physics, including electromagnetism, relativistic celestial mechanics and geodesy, as well the study gravitational radiation lensing. Their use allows integration relevant wave propagation equations to arbitrary order. We present an improved iterative method for decomposition symmetric rank. The can be both coordinate-free symbolic derivati...

We describe the structure of a free loop of rank n in the variety of automorphic Moufang loops as a subdirect product of a free group and a free commutative Moufang loop, both of rank n. In particular, the variety of automorphic Moufang loops is the join of the variety of groups and the variety of commutative Moufang loops.