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The relation showing that the Grunwald-Letnikov and generalised Cauchy derivatives are equal is presented. This establishes a bridge between two different formulations and simultaneously between the classic integer order derivatives and the fractional ones. Starting from the generalised Cauchy derivative formula, new relations are obtained, namely a regularised version that makes the concept of pseudo-function appear naturally without the need for a rejection of any infinite part. From the regularised derivative, new formulations are deduced and specialised first for the real functions and afterwards for functions with Laplace transforms obtaining the definitions proposed by Lionville. With these tools suitable definitions of fractional linear systems are obtained.

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Some results presented in the paper ?Modeling fractional stochastic systems as non-random fractional dynamics driven Brownian motions? ?I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999? are discussed in this paper. The slightly modified Grünwald-Letnikov derivative proposed there is used to deduce some interesting results that are in contradiction with those proposed in the referred paper.

Some results presented in the paper ?Modeling fractional stochastic systems as non-random fractional dynamics driven Brownian motions? ?I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999? are discussed in this paper. The slightly modified Grünwald-Letnikov derivative proposed there is used to deduce some interesting results that are in contradiction with those proposed in the referred paper. Keywords: Fractional calculus; Grünwald-Letnikov derivative; Fractional Brownian motion

The well-known Liouville, Riemann?Liouville and Caputo derivatives are extended to the complex functions space, in a natural way, and it is established interesting connections between them and the Grünwald?Letnikov derivative. Particularly, starting from a complex formulation of the Grünwald?Letnikov derivative we establishes a bridge with existing integral formulations and obtained regularised integrals for Liouville, Riemann?Liouville, and Caputo derivatives. Moreover, it is shown that we can combine the procedures followed in the computation of Riemann?Liouville and Caputo derivatives with the Grünwald?Letnikov to obtain a new way of computing them. The theory we present here will surely open a new way into the fractional derivatives computation.

Sleep staging is a crucial step before the scoring the sleep apnoea, in subjects that are tested for this condition. These patients undergo a whole night polysomnography recording that includes EEG, EOG, ECG, EMG and respiratory signals. Sleep staging refers to the quantification of its depth. Despite the commercial sleep software being able to stage the sleep, there is a general lack of confidence amongst health practitioners of these machine results. Generally the sleep scoring is done over the visual inspection of the overnight patient EEG recording, which takes the attention of an expert medical practitioner over a couple of hours. This contributes to a waiting list of two years for patients of the Portuguese Health Service. In this work we have used a spectral comparison method called Itakura distance to be able to make a distinction between sleepy and awake epochs in a night EEG recording, therefore automatically doing the staging. We have used the data from 20 patients of Hospital Pulido Valente, which had been previously visually expert scored. Our technique results were promising, in a way that Itakura distance can, by itself, distinguish with a good degree of certainty the N2, N3 and awake states. Pre-processing stages for artefact reduction and baseline removal using Wavelets were applied.