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Essential tremor (ET) is the most common movement disorder. In fact, its prevalence is about 20 times higher than that of Parkinson's disease. In addition, studies have shown that a high percentage of cases, between 50 and 70%, are estimated to be of genetic origin. The gold standard test for diagnosis, monitoring and to differentiate between both pathologies is based on the drawing of the Archimedes' spiral. Our major challenge is to develop the simplest system able to correctly classify Archimedes' spirals, therefore we will exclusively use the information of the x and y coordinates. This is the minimum information provided by any digitizing device. We explore the use of features from drawings related to the Discrete Cosine Transform as part of a wider cross-study for the diagnosis of essential tremor held at Biodonostia. We compare the performance of these features against other classic and already analyzed ones. We outperform previous results using a very simple system and a reduced set of features. Because the system is simple, it will be possible to implement it in a portable device (microcontroller), which will receive the x and y coordinates and will issue the classification result. This can be done in real time, and therefore without needing any extra job from the medical team. In future works these new drawing-biomarkers will be integrated with the ones obtained in the previous Biodonostia study. Undoubtedly, the use of this technology and user-friendly tools based on indirect measures could provide remarkable social and economic benefits. We thank the Ministry of Business and Knowledge of the Government of Catalonia that partially supported this study through the Industrial Doctorates Plan to IA-E. We also thank the grant of Domus Vi Foundation "Kms para recordar," the programs of Basque Government, ETORTEK and IT115-16, the Gipuzkoa Goverment, Red Guipuzcoana de Ciencia, Tecnologia e Innovacion, and the Ministry of Science and Innovation for the TEC2016-77791-C04-R grant, which partially supported the study. Finally we would like to thank reviewers for their detailed and helpful comments to the manuscript.

Several ways of interleaving, as studied in theoretical computer science, and some subjects from mathematics can be modeled by length-preserving operations on strings, that only permute the symbol positions in strings. Each such operation $X$ gives rise to a family $\{X_n\}_{n\geq2}$ of similar permutations. We call an integer $n$ $X$-{\em prime} if $X_n$ consists of a single cycle of length $n$ ($n\geq2$). For some instances of $X$ ---such as shuffle, twist, operations based on the Archimedes' spiral and on the Josephus problem--- we investigate the distribution of $X$-primes and of the associated (ordinary) prime numbers, which leads to variations of some well-known conjectures in number theory.

Several ways of interleaving, as studied in theoretical computer science, and some subjects from mathematics can be modeled by length-preserving operations on strings, that only permute the symbol positions in strings. Each such operation $X$ gives rise to a family $\{X_n\}_{n\geq2}$ of similar permutations. We call an integer $n$ $X$-{\em prime} if $X_n$ consists of a single cycle of length $n$ ($n\geq2$). For some instances of $X$ ---such as shuffle, twist, operations based on the Archimedes' spiral and on the Josephus problem--- we investigate the distribution of $X$-primes and of the associated (ordinary) prime numbers, which leads to variations of some well-known conjectures in number theory.

This paper is an extended version of one paper published in the 4th IEEE International Work Conference on Bioinspired Intelligence, Donostia, Spain, 9–12 June 2015 Biomedical systems produce biosignals that arise from interaction mechanisms. In a general form, those mechanisms occur across multiple scales, both spatial and temporal, and contain linear and non-linear information. In this framework, entropy measures are good candidates in order provide useful evidence about disorder in the system, lack of information in time-series and/or irregularity of the signals. The most common movement disorder is essential tremor (ET), which occurs 20 times more than Parkinson's disease. Interestingly, about 50%-70% of the cases of ET have a genetic origin. One of the most used standard tests for clinical diagnosis of ET is Archimedes' spiral drawing. This work focuses on the selection of non-linear biomarkers from such drawings and handwriting, and it is part of a wider cross study on the diagnosis of essential tremor, where our piece of research presents the selection of entropy features for early ET diagnosis. Classic entropy features are compared with features based on permutation entropy. Automatic analysis system settled on several Machine Learning paradigms is performed, while automatic features selection is implemented by means of ANOVA (analysis of variance) test. The obtained results for early detection are promising and appear applicable to real environments. This work has been partially supported by the University of the Basque Country under project ref. UPV/EHU-58/14, SAIOTEK program and others from the Basque Government, the Spanish Ministerio de Ciencia e Innovacion TEC2012-38630-C04-03, the University of Vic-Central University of Catalonia under the research grant R0904, INNPACTO program from the Spanish Government, and UPV/EHU Summer Courses Foundation.