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Abstract: Given a graph , a defensive alliance of is a set of vertices satisfying the condition that for each , at least half of the vertices in the closed neighborhood of are in . A defensive alliance is called global if every vertex in is adjacent to at least one member of the defensive alliance . The global defensive alliance number of , denoted , is the minimum size around all the global defensive alliances of . In this paper, we present an efficient algorithm to determine the global defensive alliance numbers of trees, and further give formulas to decide the global defensive alliance numbers of complete -ary trees for . We also establish upper bounds and lower bounds for and , and show that the bounds are sharp for certain . [Copyright &y& Elsevier]

Background Prognostic uncertainty inhibits clinicians from initiating timely end-of-life discussions and advance care planning. This study evaluates the efficacy of the CriSTAL ( Cri teria for S creening and T riaging to A ppropriate a L ternative care) checklist in emergency departments. Methods Prospective cohort study of patients aged ≥65 years with any diagnosis admitted via emergency departments in ten hospitals in Australia, Denmark and Ireland. Electronic and paper clinical records will be used to extract risk factors such as nursing home residency, physiological deterioration warranting a rapid response call, personal history of active chronic disease, history of hospitalisations or intensive care unit admission in the past year, evidence of proteinuria or ECG abnormalities, and evidence of frailty to be concurrently measured with Fried Score and Clinical Frailty Scale. Patients or their informal caregivers will be contacted by telephone around three months after initial assessment to ascertain survival, self-reported health, post-discharge frailty and health service utilisation since discharge. Logistic regression and bootstrapping techniques and AUROC curves will be used to test the predictive accuracy of CriSTAL for death within 90 days of admission and in-hospital death. Discussion The CriSTAL checklist is an objective and practical tool for use in emergency departments among older patients to determine individual probability of death in the short-term. Its validation in this cohort is expected to reduce clinicians’ prognostic uncertainty on the time to patients’ death and encourage timely end-of-life conversations to support clinical decisions with older frail patients and their families about their imminent or future care choices. [ABSTRACT FROM AUTHOR]

Objective: QRS detection algorithms are needed to analyze electrocardiogram (ECG) recordings generated in telehealth environments. However, the numerous published QRS detectors focus on clean clinical data. Here, a “UNSW” QRS detection algorithm is described that is suitable for clinical ECG and also poorer quality telehealth ECG. Methods: The UNSW algorithm generates a feature signal containing information about ECG amplitude and derivative, which is filtered according to its frequency content and an adaptive threshold is applied. The algorithm was tested on clinical and telehealth ECG and the QRS detection performance is compared to the Pan–Tompkins (PT) and Gutiérrez–Rivas (GR) algorithm. Results: For the MIT-BIH Arrhythmia database (virtually artifact free, clinical ECG), the overall sensitivity (Se) and positive predictivity (+P) of the UNSW algorithm was >99%, which was comparable to PT and GR. When applied to the MIT-BIH noise stress test database (clinical ECG with added calibrated noise) after artifact masking, all three algorithms had overall Se >99%, and the UNSW algorithm had higher +P (98%, p < 0.05) than PT and GR. For 250 telehealth ECG records (unsupervised recordings; dry metal electrodes), the UNSW algorithm had 98% Se and 95% +P which was superior to PT (+P: p < 0.001) and GR (Se and +P: p < 0.001). Conclusion: This is the first study to describe a QRS detection algorithm for telehealth data and evaluate it on clinical and telehealth ECG with superior results to published algorithms. Significance: The UNSW algorithm could be used to manage increasing telehealth ECG analysis workloads. [ABSTRACT FROM AUTHOR]