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Aims: It is increasingly recognized that the presence of comorbidities substantially contributes to the disease burden in patients with heart failure (HF). Several reports have suggested that clustering of comorbidities can lead to improved characterization of the disease phenotypes, which may influence management of the individual patient. Therefore, we aimed to cluster patients with HF based on medical comorbidities and their treatment and, subsequently, compare the clinical characteristics between these clusters. Methods and results: A total of 603 patients with HF entering an outpatient HF rehabilitation programme were included [median age 65 years (interquartile range 56–71), 57% ischaemic origin of cardiomyopathy, and left ventricular ejection fraction 35% (26–45)]. Exercise performance, daily life activities, disease‐specific health status, coping styles, and personality traits were assessed. In addition, the presence of 12 clinically relevant comorbidities was recorded, based on targeted diagnostics combined with applicable pharmacotherapies. Self‐organizing maps (SOMs; www.viscovery.net) were used to visualize clusters, generated by using a hybrid algorithm that applies the classical hierarchical cluster method of Ward on top of the SOM topology. Five clusters were identified: (1) a least comorbidities cluster; (2) a cachectic/implosive cluster; (3) a metabolic diabetes cluster; (4) a metabolic renal cluster; and (5) a psychologic cluster. Exercise performance, daily life activities, disease‐specific health status, coping styles, personality traits, and number of comorbidities were significantly different between these clusters. Conclusions: Distinct combinations of comorbidities could be identified in patients with HF. Therapy may be tailored based on these clusters as next step towards precision medicine. The effect of such an approach needs to be prospectively tested. [ABSTRACT FROM AUTHOR]

We study convergence of return- and hitting-time distributions of small sets $E_{k}$ with $\unicode[STIX]{x1D707}(E_{k})\rightarrow 0$ in recurrent ergodic dynamical systems preserving an infinite measure $\unicode[STIX]{x1D707}$. Some properties which are easy in finite measure situations break down in this null-recurrent set-up. However, in the presence of a uniform set $Y$ with wandering rate regularly varying of index $1-\unicode[STIX]{x1D6FC}$ with $\unicode[STIX]{x1D6FC}\in (0,1]$ , there is a scaling function suitable for all subsets of $Y$. In this case, we show that return distributions for the $E_{k}$ converge if and only if the corresponding hitting-time distributions do, and we derive an explicit relation between the two limit laws. Some consequences of this result are discussed. In particular, this leads to improved sufficient conditions for convergence to ${\mathcal{E}}^{1/\unicode[STIX]{x1D6FC}}{\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$ , where ${\mathcal{E}}$ and ${\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$ are independent random variables, with ${\mathcal{E}}$ exponentially distributed and ${\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$ following the one-sided stable law of order $\unicode[STIX]{x1D6FC}$ (and ${\mathcal{G}}_{1}:=1$). The same principle also reveals the limit laws (different from the above) which occur at hyperbolic periodic points of prototypical null-recurrent interval maps. We also derive similar results for the barely recurrent $\unicode[STIX]{x1D6FC}=0$ case. [ABSTRACT FROM AUTHOR]