A ventilator-dependent child had been in the paediatric intensive care unit (PICU) ever since birth. As a result, she had fallen behind considerably in her development. After 18 months, continuous positive airway pressure was successfully administered via a tracheostomy tube with a novel lightweight device. This enabled her to walk in the PICU. With this device, the child was discharged home wh...

Investigating the direct integral decomposition of von Neumann algebras of bounded module operators on self-dual Hilbert W*-modules an equivalence principle is obtained which connects the theory of direct disintegration of von Neumann algebras on separable Hilbert spaces and the theory of von Neumann representations on self-dual Hilbert A-modules with countably generated Apre-dual Hilbert A-mod...

These are notes from two lectures given in MATH 27200, Basic Functional Analysis, at the University of Chicago in March 2010. The proof of the spectral theorem for compact operators comes from [Zim90, Chapter 3]. 1. The Spectral Theorem for Compact Operators The idea of the proof of the spectral theorem for compact self-adjoint operators on a Hilbert space is very similar to the finite-dimensio...

We study homogenization by Γ-convergence of functionals of type Z Ω W “ x ε ,∇φ(x) ” dx, where Ω ⊂ RN is a bounded open set, φ ∈ W 1,p(Ω; Rm) and p > 1, when the 1-periodic integrand W : RN ×Mm×N → [0, +∞] is not of p-polynomial growth. Our homogenization technique can be applied when m = N and W has a singular behavior of type W (x, ξ) → +∞ as detξ → 0. However, our technique is not consistent...

It is shown that the Szegő projection S of a smoothly bounded domain Ω, not necessarily pseudoconvex, satisfies local regularity estimates at certain boundary points, provided that condition R holds for Ω. It is also shown that any biholomorphic mapping f : Ω → D between smoothly bounded domains extends smoothly near such points, provided that a weak regularity assumption holds for D. 1. Prelim...

We prove that the Dirichlet fundamental polyhedron for a cyclic group generated by a unipotent or hyperbolic element y acting on complex hyperbolic «-space centered at an arbitrary point w is bounded by the two hypersurfaces equidistant from the pairs w, yw and w,y~lw respectively. The proof relies on a convexity property of the distance to an isometric flow containing y.

The variance of the particle number (equivalently the total charge) in a domain of length L of a one-component plasma (OCP) on a cylinder of circumference W at the reciprocal temperature ;=2, is shown to remain bounded as L . This exactly solvable system with average density \ has a density profile which is periodic with period (\W ) along the axis of the infinitely long cylinder. This illustra...

Denote by W (A) the numerical range of a bounded linear operator A. For two operators A and B (which may act on different Hilbert spaces), we study the relation between the inclusion relation W (A) ⊆W (B) and the condition that A can be dilated to an operator of the form B ⊗ I. We also investigate the possibilities of dilating an operator A to operators with simple structure under the assumptio...

Let A be a compact d-rectifiable set embedded in Euclidean space R, d ≤ p. For a given continuous distribution σ(x) with respect to d-dimensional Hausdorff measure on A, our earlier results provided a method for generating N -point configurations on A that have asymptotic distribution σ(x) as N →∞; moreover such configurations are “quasi-uniform” in the sense that the ratio of the covering radi...

If D is a domain in the complex s-plane, then the family B=B(D) of bounded analytic functions in D is defined as consisting of those analytic functions ƒ(z) which are regular and single-valued in D and which satisfy the inequality \f(z) | < 1 at all points of D. The classical investigations of the family B(D) were restricted to the case in which D is a simply-connected domain. In fact, D was ge...