Hopf algebras and Markov chains: Two examples and a theory

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes will only appear on the version on Amy Pang's website, the arXiv version will not be updated.

Global Impact of the COVID-19 Pandemic on Stroke Volumes and Cerebrovascular Events: One-Year Follow-up.

Declines in stroke admission, intravenous thrombolysis, and mechanical thrombectomy volumes were reported during the first wave of the COVID-19 pandemic. There is a paucity of data on the longer-term effect of the pandemic on stroke volumes over the course of a year and through the second wave of the pandemic. We sought to measure the impact of the COVID-19 pandemic on the volumes of stroke admissions, intracranial hemorrhage (ICH), intravenous thrombolysis (IVT), and mechanical thrombectomy over a one-year period at the onset of the pandemic (March 1, 2020, to February 28, 2021) compared with the immediately preceding year (March 1, 2019, to February 29, 2020). We conducted a longitudinal retrospective study across 6 continents, 56 countries, and 275 stroke centers. We collected volume data for COVID-19 admissions and 4 stroke metrics: ischemic stroke admissions, ICH admissions, intravenous thrombolysis treatments, and mechanical thrombectomy procedures. Diagnoses were identified by their ICD-10 codes or classifications in stroke databases. There were 148,895 stroke admissions in the one-year immediately before compared to 138,453 admissions during the one-year pandemic, representing a 7% decline (95% confidence interval [95% CI 7.1, 6.9]; p<0.0001). ICH volumes declined from 29,585 to 28,156 (4.8%, [5.1, 4.6]; p<0.0001) and IVT volume from 24,584 to 23,077 (6.1%, [6.4, 5.8]; p<0.0001). Larger declines were observed at high volume compared to low volume centers (all p<0.0001). There was no significant change in mechanical thrombectomy volumes (0.7%, [0.6,0.9]; p=0.49). Stroke was diagnosed in 1.3% [1.31,1.38] of 406,792 COVID-19 hospitalizations. SARS-CoV-2 infection was present in 2.9% ([2.82,2.97], 5,656/195,539) of all stroke hospitalizations. There was a global decline and shift to lower volume centers of stroke admission volumes, ICH volumes, and IVT volumes during the 1st year of the COVID-19 pandemic compared to the prior year. Mechanical thrombectomy volumes were preserved. These results suggest preservation in the stroke care of higher severity of disease through the first pandemic year. This study is registered under NCT04934020