On Bernoulli Decompositions for Random Variables, Concentration Bounds, and Spectral Localization

As was noted already by A. N. Kolmogorov, any random variable has a Bernoulli component. This observation provides a tool for the extension of results which are known for Bernoulli random variables to arbitrary distributions. Two applications are provided here: i. an anti-concentration bound for a class of functions of independent random variables, where probabilistic bounds are extracted from combinatorial results, and ii. a proof, based on the Bernoulli case, of spectral localization for random Schroedinger operators with arbitrary probability distributions for the single site coupling constants. For a general random variable, the Bernoulli component may be defined so that its conditional variance is uniformly positive. The natural maximization problem is an optimal transport question which is also addressed here

Chains of infinite order, chains with memory of variable length, and maps of the interval

We show how to construct a topological Markov map of the interval whose invariant probability measure is the stationary law of a given stochastic chain of infinite order. In particular we caracterize the maps corresponding to stochastic chains with memory of variable length. The problem treated here is the converse of the classical construction of the Gibbs formalism for Markov expanding maps of the interval

Exponential Mixing for a Stochastic PDE Driven by Degenerate Noise

We study stochastic partial differential equations of the reaction-diffusion type. We show that, even if the forcing is very degenerate (i.e. has not full rank), one has exponential convergence towards the invariant measure. The convergence takes place in the topology induced by a weighted variation norm and uses a kind of (uniform) Doeblin condition.Comment: 10 pages, 1 figur

Estimation of total collagen volume: a T1 mapping versus histological comparison study in healthy Landrace pigs

Right ventricular biopsy represents the gold standard for the assessment of myocardial fibrosis and collagen content. This invasive technique, however, is accompanied by perioperative complications and poor reproducibility. Extracellular volume (ECV) measured through cardiovascular magnetic resonance (CMR) has emerged as a valid surrogate method to assess fibrosis non-invasively. Nonetheless, ECV provides an overestimation of collagen concentration since it also considers interstitial space. Our study aims to investigate the feasibility of estimating total collagen volume (TCV) through CMR by comparing it with the TCV measured at histology. Seven healthy Landrace pigs were acutely instrumented closed-chest and transported to the MRI facility for measurements. For each protocol, CMR imaging at 3T was acquired. MEDIS software was used to analyze T1 mapping and ECV for both the left ventricular myocardium (LVmyo) and left ventricular septum (LVseptum). ECV was then used to estimate TCVCMR at LVmyo and LVseptum following previously published formulas. Tissues were prepared following an established protocol and stained with picrosirius red to analyze the TCVhisto in LVmyo and LVseptum. TCV measured at LVmyo and LVseptum with both histology (8 ± 5 ml and 7 ± 3 ml, respectively) and T1-Mapping (9 ± 5 ml and 8 ± 6 ml, respectively) did not show any regional differences. TCVhisto and TCVCMR showed a good level of data agreement by Bland–Altman analysis. Estimation of TCV through CMR may be a promising way to non-invasively assess myocardial collagen content and may be useful to track disease progression or treatment response

On conformal measures and harmonic functions for group extensions

We prove a Perron-Frobenius-Ruelle theorem for group extensions of topological Markov chains based on a construction of $\sigma$-finite conformal measures and give applications to the construction of harmonic functions.Comment: To appear in Proceedings of "New Trends in Onedimensional Dynamics, celebrating the 70th birthday of Welington de Melo

Brief research report: Quantitative analysis of potential coronary microvascular disease in suspected long-COVID syndrome

BACKGROUND: Case series have reported persistent cardiopulmonary symptoms, often termed long-COVID or post-COVID syndrome, in more than half of patients recovering from Coronavirus Disease 19 (COVID-19). Recently, alterations in microvascular perfusion have been proposed as a possible pathomechanism in long-COVID syndrome. We examined whether microvascular perfusion, measured by quantitative stress perfusion cardiac magnetic resonance (CMR), is impaired in patients with persistent cardiac symptoms post-COVID-19. METHODS: Our population consisted of 33 patients post-COVID-19 examined in Berlin and London, 11 (33%) of which complained of persistent chest pain and 13 (39%) of dyspnea. The scan protocol included standard cardiac imaging and dual-sequence quantitative stress perfusion. Standard parameters were compared to 17 healthy controls from our institution. Quantitative perfusion was compared to published values of healthy controls. RESULTS: The stress myocardial blood flow (MBF) was significantly lower [31.8 ± 5.1 vs. 37.8 ± 6.0 (μl/g/beat), P < 0.001] and the T2 relaxation time was significantly higher (46.2 ± 3.6 vs. 42.7 ± 2.8 ms, P = 0.002) post-COVID-19 compared to healthy controls. Stress MBF and T1 and T2 relaxation times were not correlated to the COVID-19 severity (Spearman r = −0.302, −0.070, and −0.297, respectively) or the presence of symptoms. The stress MBF showed a U-shaped relation to time from PCR to CMR, no correlation to T1 relaxation time, and a negative correlation to T2 relaxation time (Pearson r = −0.446, P = 0.029). CONCLUSION: While we found a significantly reduced microvascular perfusion post-COVID-19 compared to healthy controls, this reduction was not related to symptoms or COVID-19 severity