Let (Fn)n∈N be a sequence of non-decreasing functions from [0,+∞) into [0,+∞). Under some suitable hypotheses on (Fn)n∈N, we prove that if g ∈ Lp(RN ), 1 < p < +∞, satisfies lim inf n→∞ ∫ RN ∫ RN Fn(|g(x)− g(y)|) |x − y|N+p dx dy < +∞, then g ∈ W1,p(RN ) and moreover lim n→∞ ∫ RN ∫ RN Fn(|g(x)− g(y)|) |x − y|N+p dx dy = KN,p ∫ RN |∇g(x)| dx, whereKN,p is a positive constant depending only onN a...

1. Overview 2 2. Electromagnetic waves 3 2.1. Approximate cloaking via change of variables 3 2.1.1. Cloaking for the Helmholtz equation 3 2.1.2. Full range estimates for the degree of invisibility 4 2.1.3. Perfect cloaking for the Helmholtz equation 4 2.2. Electrical impedance tomography 5 2.3. Generalized impedance boundary conditions 5 2.3.1. The Helmholtz equation 6 2.3.1. The time harmonic ...

In this paper, we establish the equivalence between the Haj lasz-Sobolev spaces or classical Triebel-Lizorkin spaces and a class of grand Triebel-Lizorkin spaces on Euclidean spaces and also on metric spaces that are both doubling and reverse doubling. In particular, when p ∈ (n/(n + 1),∞), we give a new characterization of the Haj laszSobolev spaces Ṁ (R) via a grand Littlewood-Paley function.

It is known that the generating vector of a rank-1 lattice rule can be constructed component-by-component to achieve strong tractability error bounds in both weighted Korobov spaces and weighted Sobolev spaces. Since the weights for these spaces are nonincreasing, the first few variables are in a sense more important than the rest. We thus propose to copy the points of a rank-1 lattice rule a n...