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Funding Information: This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UIDB/00297/2020 (Centro de Matemática e Aplicações). Publisher Copyright: © 2021, Springer Nature Switzerland AG. The synchronization phenomenon was reported for the first time by Christiaan Huygens, when he noticed the strange tendency of a couple of clocks to synchronise their movements. More recently this phenomena was shown to be ubiquitous in nature and it is broadly studied by its applications, for example in biological cycles. We consider the problem of synchronization of a general network of linearly coupled oscillators, not necessarily identical. In this case the existence of a linear synchronization space is not expected, so we present an approach based on the proof of the existence of a synchronization manifold, the so-called generalised synchronization. Based on some results developed by R. Smith and on Wazewski’s principle, a general theory on the existence of invariant manifolds that attract the solutions of the system that are bounded in the future, is presented. Applications and estimates on parameters for the existence of synchronization are presented for several examples: systems of coupled pendulum type equations, coupled Lorenz systems of equations, and oscillators coupled through a medium, among many others. authorsversion published

Work partially supported by the Portuguese Science Foundation, FCT, through the projects The Notion of Mathematical Proof, PTDC/MHC-FIL/5363/2012, Hilbert’s 24th Problem, PTDC/MHC-FIL/2583/2014, and the Centro de Matemática e Aplicações, UID/MAT/00297/2013. This paper provides a discussion to which extent the Mathematician David Hilbert could or should be considered as a Philosopher, too. In the first part, we discuss some aspects of the relation of Mathematicians and Philosophers. In the second part we give an analysis of David Hilbert as Philosopher. authorsversion published

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.