We study the “smallness” of the set of non-hypercyclic vectors for some classical hypercyclic operators.

We give a condition for the periodic, three dimensional, incompressible Navier-Stokes equations to be globally wellposed. This condition is not a smallness condition on the initial data, as the data is allowed to be arbitrarily large in the scale invariant space B ∞,∞, which contains all the known spaces in which there is a global solution for small data. The smallness condition is rather a non...

We analyze the horizon structure of families of space times obtained by evolving initial data sets containing apparent horizons with several connected components. We show that under certain smallness conditions the outermost apparent horizons will also have several connected components. We further show that, again under a smallness condition, the maximal globally hyperbolic development of the m...

Several maternal and fetal physiological characteristics account for a substantial proportion of the variation in birth weight. These characteristics can be used to calculate an individualized optimal birth weight and to adjust or 'customize' the birth weight standard. Customized birth weight standards improve the distinction between constitutional and pathological smallness, and there is evide...

We show that minimizers of free discontinuity problems with energy dependent on jump integrals and Dirichlet boundary conditions are smooth provided a smallness condition is imposed on data. We examine several examples, including elastic-plastic beam and plate with free yield lines and deformable body with free damage. In all cases there is a gap between the condition for solvability (safe load...