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.

Association of interatrial septal abnormalities with cardiac impulse conduction disorders in adult patients: experience from a tertiary center in Kosovo

Interatrial septal disorders, which include: atrial septal defect, patent foramen ovale and atrial septal aneurysm, are frequent congenital anomalies found in adult patients. Early detection of these anomalies is important to prevent their hemodynamic and/or thromboembolic consequences. The aims of this study were: to assess the association between impulse conduction disorders and anomalies of interatrial septum; to determine the prevalence of different types of interatrial septum abnormalities; to assess anatomic, hemodynamic, and clinical consequences of interatrial septal pathologies. Fifty-three adult patients with impulse conduction disorders and patients without ECG changes but with signs of interatrial septal abnormalities, who were referred to our center for echocardiography, were included in a prospective transesophageal echocardiography study. Interatrial septal anomalies were detected in around 85% of the examined patients. Patent foramen ovale was encountered in 32% of the patients, and in combination with atrial septal aneurysm in an additional 11.3% of cases. Atrial septal aneurysm and atrial septal defect were diagnosed with equal frequency in 20.7% of our study population. Impulse conduction disorders were significantly more suggestive of interatrial septal anomalies than clinical signs and symptoms observed in our patients (84.91% vs 30.19%, P=0.002). Right bundle branch block was the most frequent impulse conduction disorder, found in 41 (77.36%) cases. We conclude that interatrial septal anomalies are highly associated with impulse conduction disorders, particularly with right bundle branch block. Impulse conduction disorders are more indicative of interatrial septal abnormalities in earlier stages than can be understood from the patient’s clinical condition

Correlation of pulse wave velocity with left ventricular mass in patients with hypertension once blood pressure has been normalized

Vascular stiffness has been proposed as a simple method to assess arterial loading conditions of the heart which induce left ventricular hypertrophy (LVH). There is some controversy as to whether the relationship of vascular stiffness to LVH is independent of blood pressure, and which measurement of arterial stiffness, augmentation index (AI) or pulse wave velocity (PWV) is best. Carotid pulse wave contor and pulse wave velocity of patients (n=20) with hypertension whose blood pressure (BP) was under control (<140/90 mmHg) with antihypertensive drug treatment medications, and without valvular heart disease, were measured. Left ventricular mass, calculated from 2D echocardiogram, was adjusted for body size using two different methods: body surface area and height. There was a significant (P<0.05) linear correlation between LV mass index and pulse wave velocity. This was not explained by BP level or lower LV mass in women, as there was no significant difference in PWV according to gender (1140.1+67.8 vs 1110.6+57.7 cm/s). In contrast to PWV, there was no significant correlation between LV mass and AI. In summary, these data suggest that aortic vascular stiffness is an indicator of LV mass even when blood pressure is controlled to less than 140/90 mmHg in hypertensive patients. The data further suggest that PWV is a better proxy or surrogate marker for LV mass than AI and the measurement of PWV may be useful as a rapid and less expensive assessment of the presence of LVH in this patient population