Realizations of Real Low-Dimensional Lie Algebras

Using a new powerful technique based on the notion of megaideal, we construct a complete set of inequivalent realizations of real Lie algebras of dimension no greater than four in vector fields on a space of an arbitrary (finite) number of variables. Our classification amends and essentially generalizes earlier works on the subject. Known results on classification of low-dimensional real Lie algebras, their automorphisms, differentiations, ideals, subalgebras and realizations are reviewed.Comment: LaTeX2e, 39 pages. Essentially exetended version. Misprints in Appendix are correcte

Cepstrum Feature Selection for the Classification of Sleep Apnea-Hypopnea Syndrome based on Heart Rate Variability

Abstract Cepstrum Coefficients are analyzed in order to study its performance in Sleep Apnea Introduction The Sleep Apnea Hypopnea Syndrome (SAHS) is a respiratory disorder characterized by frequent breathing pauses and a collapse of pharynx during sleep. If breathing ceases completely, then the event is called apnea. In case breathing does not cease but there is a reduction in the volume of air entering the lungs, then the event is called hipopnea. Previous studies have tried to diagnostic SAHS with the RR series obtained from the electrocardiogram (ECG) [1] with good performance, anyway the underlying regulatory mechanisms during apnea are still poorly understood. This fact makes necessary to explore appropriate feature estimation techniques in order to extract as much information as possible. In previous contribution [2] we have used cepstrum features without taking into consideration any selection criteria. In this paper we apply forward feature selection in order to improve apnea screening performance and find coefficients which describe with more detail the RR pattern in presence of SAHS. We have selected features from a specific cepstrum coefficients set composed by the first 60 elements containing information about periodic structures of the RR series but also about the system modelled by the filter-type elements. Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA) have been proposed in order to quantify apnea minutes. The system will provide also a global score of the presence of clinically significant apnea based on the minute by minute apnea detection. A subject will be classified globally as SAHS is the percentage of minutes with apnea is at least 16%. Database The database was provided by Prof. Dr. Thomas Penzel for Computers in Cardiology 2000 challenge [3]. The data have been divided divided into a learning set (L set) and a test set (T set) of equal size. Each set consists of 35 recordings, containing a single ECG signal digitized at 100 Hz with 12-bit resolution, continuously for approximately 8 hours. Each recording includes a set of reference annotations, one for each minute, which indicates the presence or absence of apnea during that minute. These reference annotations were made by human experts on the basis of simultaneously recorded respiration signals. Group A (apnea) contains recordings with at least 10