Uniqueness for the signature of a path of bounded variation and the reduced path group

We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is one special tree reduced path. The set of these paths is the Reduced Path Group. It is a continuous analogue to the group of reduced words. The signature of the path is a power series whose coefficients are definite iterated integrals of the path. We identify the paths with trivial signature as the tree-like paths, and prove that two paths are in tree-like equivalence if and only if they have the same signature. In this way, we extend Chen's theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for reparameterisation in the general setting. It is suggestive to think of this result as a non-commutative analogue of the result that integrable functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients. As a second theme we give quantitative versions of Chen's theorem in the case of lattice paths and paths with continuous derivative, and as a corollary derive results on the triviality of exponential products in the tensor algebra.Comment: 52 pages - considerably extended and revised version of the previous version of the pape

Determinantal probability measures

Determinantal point processes have arisen in diverse settings in recent years and have been investigated intensively. We study basic combinatorial and probabilistic aspects in the discrete case. Our main results concern relationships with matroids, stochastic domination, negative association, completeness for infinite matroids, tail triviality, and a method for extension of results from orthogonal projections to positive contractions. We also present several new avenues for further investigation, involving Hilbert spaces, combinatorics, homology, and group representations, among other areas.Comment: 50 pp; added reference to revision. Revised introduction and made other small change

Distance covariance in metric spaces

We extend the theory of distance (Brownian) covariance from Euclidean spaces, where it was introduced by Sz\'{e}kely, Rizzo and Bakirov, to general metric spaces. We show that for testing independence, it is necessary and sufficient that the metric space be of strong negative type. In particular, we show that this holds for separable Hilbert spaces, which answers a question of Kosorok. Instead of the manipulations of Fourier transforms used in the original work, we use elementary inequalities for metric spaces and embeddings in Hilbert spaces.Comment: Published in at http://dx.doi.org/10.1214/12-AOP803 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The 2-component dispersionless Burgers equation arising in the modelling of blood flow

This article investigates the properties of the solutions of the dispersionless two-component Burgers (B2) equation, derived as a model for blood-flow in arteries with elastic walls. The phenomenon of wave breaking is investigated as well as applications of the model to clinical conditions.Comment: 13 pages, 2 figures. arXiv admin note: text overlap with arXiv:1009.5374 by other author

Will the New U.K. Competition and Markets Authority Make Better Antitrust Decisions?

The United Kingdom has a unique set of institutions charged with enforcing competition law. The twin pillars are the Competition Commission (“CC”) and the Office of Fair Trading (“OFT”). In the coming parliament, legislation will be passed to merge them into a new Competition and Markets Authority (“CMA”), probably with effect from 2014.2 They each have a high reputation and are regularly ranked alongside the DOJ, FTC, and DG Competition as among the best in the world. OK, few would argue that any of these institutions is unimprovable, but it does mean there is much that could be lost if the CMA is less effective than its predecessors. Should we be worried