Let $Gamma$ be a group, $Gamma'$ a subgroup of $Gamma$ with finite index and $M$ be a $Gamma$-module. We show that $M$ is cotorsion if and only if it is cotorsion as a $Gamma'$-module. Using this result, we prove that the global cotorsion dimensions of rings $ZGamma$ and $ZGamma'$ are equal.

Using Voiculescu’s notion of a matricial microstate we introduce fractal dimensions and entropy for finite sets of self-adjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of free Hausdorff dimension is an algebrai...

Given a set of classifiers and a probability distribution over their domain, one can define a metric by taking the distance between a pair of classifiers to be the probability that they classify a random item differently. We prove bounds on the sample complexity of PAC learning in terms of the doubling dimension of this metric. These bounds imply known bounds on the sample complexity of learnin...

Using Voiculescu’s notion of a matricial microstate we introduce fractal dimensions and entropies for finite sets of selfadjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of free Hausdorff dimension is an algebra...

Multi-fractal patterns occur widely in nature. In developing new algorithms to determine multi-fractal spectra of experimental data I am lead to the conclusion that generalised dimensions Dq of order q ≤ 0, including the Hausdorff dimension, are effectively irrelevant. The reason is that these dimensions are extraordinarily sensitive to regions of low density in the multi-fractal data. Instead,...