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Further applications of the Cauchon algorithm to rank determination and bidiagonal factorization
(2018)
For a class of matrices connected with Cauchon diagrams, Cauchon matrices, and the Cauchon algorithm, a method for determining the rank, and for checking a set of consecutive row (or column) vectors for linear independence is presented. Cauchon diagrams are also linked to the elementary bidiagonal factorization of a matrix and to certain types of rank conditions associated with submatrices called descending rank conditions.
The class of square matrices of order n having a negative determinant and all their minors up to order n-1 nonnegative is considered. A characterization of these matrices is presented which provides an easy test based on the Cauchon algorithm for their recognition. Furthermore, the maximum allowable perturbation of the entry in position (2,2) such that the perturbed matrix remains in this class is given. Finally, it is shown that all matrices lying between two matrices of this class with respect to the checkerboard ordering are contained in this class, too.
In 1970, B.A. Asner, Jr., proved that for a real quasi-stable polynomial, i.e., a polynomial whose zeros lie in the closed left half-plane of the complex plane, its finite Hurwitz matrix is totally nonnegative, i.e., all its minors are nonnegative, and that the converse statement is not true. In this work, we explain this phenomenon in detail, and provide necessary and sufficient conditions for a real polynomial to have a totally nonnegative finite Hurwitz matrix.
In this paper, rectangular matrices whose minors of a given order have the same strict sign are considered and sufficient conditions for their recognition are presented. The results are extended to matrices whose minors of a given order have the same sign or are allowed to vanish. A matrix A is called oscillatory if all its minors are nonnegative and there exists a positive integer k such that A^k has all its minors positive. As a generalization, a new type of matrices, called oscillatory of a specific order, is introduced and some of their properties are investigated.
Positive systems play an important role in systems and control theory and have found applications in multiagent systems, neural networks, systems biology, and more. Positive systems map the nonnegative orthant to itself (and also the non-positive orthant to itself). In other words, they map the set of vectors with zero sign variation to itself. In this article, discrete-time linear systems that map the set of vectors with up to k-1 sign variations to itself are introduced. For the special case k = 1 these reduce to discrete-time positive linear systems. Properties of these systems are analyzed using tools from the theory of sign-regular matrices. In particular, it is shown that almost every solution of such systems converges to the set of vectors with up to k-1 sign variations. It is also shown that these systems induce a positive dynamics of k-dimensional parallelotopes.
This study investigates the application of Force Sensing Resistor (FSR) sensors and machine learning algorithms for non-invasive body position monitoring during sleep. Although reliable, traditional methods like Polysomnography (PSG) are invasive and unsuited for extended home-based monitoring. Our approach utilizes FSR sensors placed beneath the mattress to detect body positions effectively. We employed machine learning techniques, specifically Random Forest (RF), K-Nearest Neighbors (KNN), and XGBoost algorithms, to analyze the sensor data. The models were trained and tested using data from a controlled study with 15 subjects assuming various sleep positions. The performance of these models was evaluated based on accuracy and confusion matrices. The results indicate XGBoost as the most effective model for this application, followed by RF and KNN, offering promising avenues for home-based sleep monitoring systems.
Cardiovascular diseases (CVD) are leading contributors to global mortality, necessitating advanced methods for vital sign monitoring. Heart Rate Variability (HRV) and Respiratory Rate, key indicators of cardiovascular health, are traditionally monitored via Electrocardiogram (ECG). However, ECG's obtrusiveness limits its practicality, prompting the exploration of Ballistocardiography (BCG) as a non-invasive alternative. BCG records the mechanical activity of the body with each heartbeat, offering a contactless method for HRV monitoring. Despite its benefits, BCG signals are susceptible to external interference and present a challenge in accurately detecting J-Peaks. This research uses advanced signal processing and deep learning techniques to overcome these limitations. Our approach integrates accelerometers for long-term BCG data collection during sleep, applying Discrete Wavelet Transforms (DWT) and Ensemble Empirical Mode Decomposition (EEMD) for feature extraction. The Bi-LSTM model, leveraging these features, enhances heartbeat detection, offering improved reliability over traditional methods. The study's findings indicate that the combined use of DWT, EEMD, and Bi-LSTM for J-Peak detection in BCG signals is effective, with potential applications in unobtrusive long-term cardiovascular monitoring. Our results suggest that this methodology could contribute to HRV monitoring, particularly in home settings, enhancing patient comfort and compliance.
Sleep is essential to physical and mental health. However, the traditional approach to sleep analysis—polysomnography (PSG)—is intrusive and expensive. Therefore, there is great interest in the development of non-contact, non-invasive, and non-intrusive sleep monitoring systems and technologies that can reliably and accurately measure cardiorespiratory parameters with minimal impact on the patient. This has led to the development of other relevant approaches, which are characterised, for example, by the fact that they allow greater freedom of movement and do not require direct contact with the body, i.e., they are non-contact. This systematic review discusses the relevant methods and technologies for non-contact monitoring of cardiorespiratory activity during sleep. Taking into account the current state of the art in non-intrusive technologies, we can identify the methods of non-intrusive monitoring of cardiac and respiratory activity, the technologies and types of sensors used, and the possible physiological parameters available for analysis. To do this, we conducted a literature review and summarised current research on the use of non-contact technologies for non-intrusive monitoring of cardiac and respiratory activity. The inclusion and exclusion criteria for the selection of publications were established prior to the start of the search. Publications were assessed using one main question and several specific questions. We obtained 3774 unique articles from four literature databases (Web of Science, IEEE Xplore, PubMed, and Scopus) and checked them for relevance, resulting in 54 articles that were analysed in a structured way using terminology. The result was 15 different types of sensors and devices (e.g., radar, temperature sensors, motion sensors, cameras) that can be installed in hospital wards and departments or in the environment. The ability to detect heart rate, respiratory rate, and sleep disorders such as apnoea was among the characteristics examined to investigate the overall effectiveness of the systems and technologies considered for cardiorespiratory monitoring. In addition, the advantages and disadvantages of the considered systems and technologies were identified by answering the identified research questions. The results obtained allow us to determine the current trends and the vector of development of medical technologies in sleep medicine for future researchers and research.
Accurate monitoring of a patient's heart rate is a key element in the medical observation and health monitoring. In particular, its importance extends to the identification of sleep-related disorders. Various methods have been established that involve sensor-based recording of physiological signals followed by automated examination and analysis. This study attempts to evaluate the efficacy of a non-invasive HR monitoring framework based on an accelerometer sensor specifically during sleep. To achieve this goal, the motion induced by thoracic movements during cardiac contractions is captured by a device installed under the mattress. Signal filtering techniques and heart rate estimation using the symlets6 wavelet are part of the implemented computational framework described in this article. Subsequent analysis indicates the potential applicability of this system in the prognostic domain, with an average error margin of approximately 3 beats per minute. The results obtained represent a promising advancement in non-invasive heart rate monitoring during sleep, with potential implications for improved diagnosis and management of cardiovascular and sleep-related disorders.