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We express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations. We completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed.

We use free probability techniques for computing spectra and Brown measures of some non hermitian operators in finite von Neumann algebras. Examples include u n +u ∞ where u n and u ∞ are the generators of Z n and Z respectively, in the free product Z n * Z, or elliptic elements, of the form S α + iS β where S α and S β are free semi-circular elements of variance α and β.

We prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution. We study the Voiculescu transform first by giving a survey of its combinatorial implications and then analytically, including a proof of free infinite divisibility. In fact we prove that a subfamily Askey-Wimp-Kerov distributions are freely infinitely divisible, of which the normal distribution is a special case.The Cauchy-Stieltjes transform of a measure µ on the real line is defined bywhen µ is a positive measure, this function maps C + into the lower half-plane. For details we refer to the excellent book of Akhieser [1]. For c = 0, we have µ 0 = γ, the normal distribution [27], and one can easily check that one can extend this family continuously to c = −1 by letting µ −1 = δ 0 , the probability giving mass one to {0}. This family, introduced in [4], plays an important role in [27]; we shall call its members the Askey-Wimp-Kerov distributions. It will turn out from our proof that {µ c : c ∈ [−1, 0]} are freely infinitely divisible. Numerical computations show that for several values of c > 0, µ c is not freely infinitely divisible. Numerical evidence seems also to indicate that µ c is classicaly infinitely divisible only when c = 0 or c = −1. An interesting interpolation between the normal and the semicircle law was constructed by Bryc, Dembo and Jiang [18] and further investigated by Buchholz [19]. This leads to a generalized Brownian motion, given by a weight function (0 < b < 1)

Abstract. Cumulants linearize convolution of measures. We use a formula of Good to define noncommutative cumulants in a very general setting. It turns out that the essential property needed is exchangeability of random variables. Roughly speaking the formula says that cumulants are moments of a certain "discrete Fourier transform" of a random variable. This provides a simple unified method to understand the known examples of cumulants, like classical, free and various q-cumulants.

Summary The present work aims to identify critical materials in water electrolysers with potential future supply constraints. The expected rise in demand for green hydrogen as well as the respective implications on material availability are assessed by conducting a case study for Germany. Furthermore, the recycling of end‐of‐life (EoL) electrolysers is evaluated concerning its potential in ensuring the sustainable supply of the considered materials. As critical materials bear the risk of raising production costs of electrolysers substantially, this article examines the readiness of this technology for industrialisation from a material perspective. Except for titanium, the indicators for each assessed material are scored with a moderate to high (platinum) or mostly high (iridium, scandium and yttrium) supply risk. Hence, the availability of these materials bears the risk of hampering the scale‐up of electrolysis capacity. Although conventional recycling pathways for platinum, iridium and titanium already exist, secondary material from EoL electrolysers will not reduce the dependence on primary resources significantly within the period under consideration—from 2020 until 2050. Notably, the materials identified as critical are used in PEM and high temperature electrolysis, whereas materials in alkaline electrolysis are not exposed to significant supply risks.

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A combinatorial formula is derived which expresses free cumulants in terms of classical cumulants. As a corollary, we give a combinatorial interpretation of free cumulants of classical distributions, notably Gaussian and Poisson distributions. The latter count connected pairings and connected partitions, respectively. The proof relies on Möbius inversion on the partition lattice.