We give some new characterizations of exactness for locally compact second countable groups. In particular, we prove that a locally compact second countable group is exact if and only if it admits a topologically amenable action on a compact Hausdorff space. This answers an open question by Anantharaman-Delaroche.

Background Heart failure is a clinical syndrome characterised by a reduced ability of the heart to pump blood. Patients with heart failure have a high mortality rate, and physicians need reliable prognostic predictions to make informed decisions about the appropriate application of devices, transplantation, medications, and palliative care. In this study, we demonstrate that combining symbolic regression with the Cox proportional hazards model improves the ability to predict death due to heart failure compared to using the Cox proportional hazards model alone. Methods We used a newly invented symbolic regression method called the QLattice to analyse a data set of medical records for 299 Pakistani patients diagnosed with heart failure. The QLattice identified non-linear mathematical transformations of the available covariates, which we then used in a Cox model to predict survival. Results An exponential function of age, the inverse of ejection fraction, and the inverse of serum creatinine were identified as the best risk factors for predicting heart failure deaths. A Cox model fitted on these transformed covariates had improved predictive performance compared with a Cox model on the same covariates without mathematical transformations. Conclusion Symbolic regression is a way to find transformations of covariates from patients’ medical records which can improve the performance of survival regression models. At the same time, these simple functions are intuitive and easy to apply in clinical settings. The direct interpretability of the simple forms may help researchers gain new insights into the actual causal pathways leading to deaths.

Background: Heart failure is a clinical syndrome characterised by a reduced ability of the heart to pump blood. Patients with heart failure have a high mortality rate, and physicians need reliable prognostic predictions to make informed decisions about the appropriate application of devices, transplantation, medications, and palliative care. In this study, we demonstrate that combining symbolic regression with the Cox proportional hazards model improves the ability to predict death due to heart failure compared to using the Cox proportional hazards model alone. Methods: We used a newly invented symbolic regression method called the QLattice to analyse a data set of medical records for 299 Pakistani patients diagnosed with heart failure. The QLattice identified a minimal set of mathematical transformations of the available covariates, which we then used in a Cox model to predict survival.Results: An exponential function of age, the inverse of ejection fraction, and the inverse of serum creatinine were identified as the best risk factors for predicting heart failure deaths. A Cox model fitted on these transformed covariates had improved predictive performance compared with a Cox model on the same covariates without mathematical transformations. Conclusion: Symbolic regression is a way to find transformations of covariates from patients’ medical records which can improve the performance of survival regression models. At the same time, these simple functions are intuitive and easy to apply in clinical settings. The direct interpretability of the simple forms may help researchers gain new insights into the actual causal pathways leading to deaths.

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Given two finitely generated groups that coarsely embed into a Hilbert space, it is known that their wreath product also embeds coarsely into a Hilbert space. We introduce a wreath product construction for general metric spaces X, Y, Z and derive a condition, called the (δ-polynomial) path lifting property, such that coarse embeddability of X, Y and Z implies coarse embeddability of X Z Y. We also give bounds on the compression of X Z Y in terms of δ and the compressions of X, Y and Z.2010 Mathematics subject classification: primary 20F65; secondary 20E22, 20F69.