We present the path-sum formulation for OE [H](t ′ , t) = T exp t ′ t H(τ ) dτ , the time-ordered exponential of a time-dependent matrix H(t). The path-sum formulation gives OE[H] as a branched continued fraction of finite depth and breadth. The terms of the path-sum have an elementary interpretation as selfavoiding walks and self-avoiding polygons on a graph. Our result is based on a representation of the time-ordered exponential as the inverse of an operator, the mapping of this inverse to sums of walks on graphs and the algebraic structure of sets of walks. We give examples demonstrating our approach. We establish a super-exponential decay bound for the magnitude of the entries of the timeordered exponential of sparse matrices. We give explicit results for matrices with commonly encountered sparse structures.Keywords: Time-ordered exponential, path-ordered exponential, path-sum, differential equation, finite continued fraction, super-exponential decay ContextThe time-ordered exponential function OE[H](t ′ , t) = T exp t ′ t H(τ ) dτ , also known as path-ordered exponential, is the unique solution of the system of differential equationssuch that OE[H](t ′ , t ′ ) = I is the identity at all times. We take H ∈ C n×n [I], n ∈ N\{0}, to be a matrix depending smoothly on the continuous variable t ∈ I, which we call time without loss of generality. In spite of the importance of the

Asthma has been associated with a higher incidence of myocardial infarction (MI), higher prevalence of MI risk factors and higher burden of cardiovascular diseases. However, detailed associations between the presentation and initial management at the time of MI and post-MI outcomes in people with asthma compared to the general population have not been studied. A total of 300,161 people were identified with a first MI over the period 2003–2013 in the Myocardial Ischaemia National Audit Project database, of whom 8922 (3%) had asthma. Logistic regression was used to compare presentation, in-hospital care, in-hospital and 180-day post-discharge all-cause mortality in people with and without asthma adjusting for demographics and comorbidities, diagnosis on arrival and secondary prevention. People with asthma were more likely to have a delay in their MI diagnosis following an STEMI (ST-elevation myocardial infarction; odds ratio (OR) 1.38, confidence interval CI 1.06–1.79) but not an nSTEMI (non-ST-elevation myocardial infarction; OR 1.04, CI 0.92–1.17) compared to people without asthma and a delay in reperfusion (OR 1.19, CI 1.09–1.30) following an STEMI. They were much less likely to be discharged on a beta blocker following an STEMI or nSTEMI (OR 0.24, CI 0.21–0.28 and OR 0.27, CI 0.24–0.30, respectively). There was no difference in in-hospital or 180-day mortality (OR 0.98, CI 0.59–1.62 and OR 0.99, CI 0.72–1.36) following an STEMI or nSTEMI (OR 0.89, CI 0.47–1.68 and OR 1.05, CI 0.85–1.28). Although people with asthma were more likely to have a delay in diagnosis following an STEMI but not an nSTEMI compared to the general population, were more likely to have a delay in reperfusion therapy and were much less likely to receive beta blockers following an STEMI or nSTEMI, there was no difference in the prescriptions of other secondary prevention medications. None of the differences in presentation or management were associated with an increase in all-cause in-hospital or 180-day mortality in people with asthma compared to the general population.