On Efficiency and Localisation for the Torsion Function

We consider the torsion function for the Dirichlet Laplacian −Δ, and for the Schrödinger operator −Δ + V on an open set ${\Omega }\subset \mathbb {R}^{m}$ Ω ⊂ ℝ m of finite Lebesgue measure $0<|{\Omega }|<\infty $ 0 < | Ω | < ∞ with a real-valued, non-negative, measurable potential V. We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction.

Exponential polynomials with Fatou and non-escaping sets of finite Lebesgue measure

We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial f both have finite Lebesgue measure. Essentially, these conditions are designed such that $|f(z)|\ge \exp (|z|^\alpha )$ for some $\alpha>0$ and all z outside a set of finite Lebesgue measure.

Three Problems on Exponential Bases

We consider three special and significant cases of the following problem. Let $D\subset \mathbb{R}^{d}$ be a (possibly unbounded) set of finite Lebesgue measure. Let $E(\mathbb{Z}^{d})=\{e^{2\unicode[STIX]{x1D70B}ix\cdot n}\}\text{}_{n\in \mathbb{Z}^{d}}$ be the standard exponential basis on the unit cube of $\mathbb{R}^{d}$. Find conditions on $D$ for which $E(\mathbb{Z}^{d})$ is a frame, a Riesz sequence, or a Riesz basis for $L^{2}(D)$.

On two functionals involving the maximum of the torsion function

In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider T(Ω)∕(M(Ω)|Ω|) and M(Ω)λ1(Ω), where Ω is a bounded open set of ℝd with finite Lebesgue measure |Ω|, M(Ω) denotes the maximum of the torsion function, T(Ω) the torsion, and λ1(Ω) the first Dirichlet eigenvalue. Particular attention is devoted to the subclass of convex sets.

The lemma of the logarithmic derivative for subharmonic functions

The classical statement of the lemma in question [7], [3] is about meromorphic functions f on ℂ and says thatfor all r > 0, with the possible exception of a set of finite Lebesgue measure. Here T(r, f) is the Nevanlinna characteristic of f. The lemma plays an important role in value distribution theory.

Osculatory Packings by Spheres

If U is an open set in Euclidean N-space EN which has finite Lebesgue measure |U| then a complete packing of U by open spheres is a collection C={Sn} of pairwise disjoint open spheres contained in U and such that Σ∞n=1|Sn| = |U|. Such packings exist by Vitali's theorem. An osculatory packing is one in which the spheres Sn are chosen recursively so that from a certain point on Sn+1 is the largest possible sphere contained in (Here S- will denote the closure of a set S). We give here a simple proof of the "well-known" fact that an osculatory packing is a complete packing. Our method of proof shows also that for osculatory packings, the Hausdorff dimension of the residual set is dominated by the exponent of convergence of the radii of the Sn.