The matrix arithmetic–geometric mean inequality revisited

The matrix arithmetic-geometric mean inequality revisited

Tusnády’s Inequality Revisited

Tusnady’s inequality is the key ingredient in the KMT/Hungarian coupling of the empirical distribution function with a Brownian Bridge. We present an elementary proof of a result that sharpens the Tusnady inequality, modulo constants. Our method uses the beta integral representation of binomial tails, simple Taylor expansion, and some novel bounds for the ratios of normal tail probabilities. Ke...

Pisier’s Inequality Revisited

Given a Banach space X, for n ∈ N and p ∈ (1,∞) we investigate the smallest constant P ∈ (0,∞) for which every f1, . . . , fn : {−1, 1} → X satisfy ∫ {−1,1}n ∥∥∥∥ n ∑ j=1 ∂jfj(ε) ∥∥∥∥ p dμ(ε) 6 P ∫ {−1,1}n ∫ {−1,1}n ∥∥∥∥ n ∑ j=1 δj∆fj(ε) ∥∥∥∥ p dμ(ε)dμ(δ), where μ is the uniform probability measure on the discrete hypercube {−1, 1} and {∂j}j=1 and ∆ = ∑n j=1 ∂j are the hypercube partial derivat...

Logarithmic Sobolev inequality revisited

r é s u m é Nous donnons une nouvelle caractérisation de l’inégalité de Sobolev logarithmique. © 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Biofilms: the matrix revisited.

Microbes often construct and live within surface-associated multicellular communities known as biofilms. The precise structure, chemistry and physiology of the biofilm all vary with the nature of its resident microbes and local environment. However, an important commonality among biofilms is that their structural integrity critically depends upon an extracellular matrix produced by their consti...