Convolution type Calderón-Zygmund singular integral operators with rough kernels p.v. Ω(x)/|x| are studied. A condition on Ω implying that the corresponding singular integrals and maximal singular integrals map L → L for 1 < p < ∞ is obtained. This condition is shown to be different from the condition Ω ∈ H1(Sn−1).

If f : [0, 1] → R is strictly increasing and continuous define Tfx = f(x) (mod 1). A transformation T̃ : [0, 1] → [0, 1] is called ε-close to Tf , if T̃ x = f̃(x) (mod 1) for a strictly increasing and continuous function f̃ : [0, 1] → R with ‖f̃ − f‖∞ < ε. It is proved that the topological pressure p(Tf , g) is lower semi-continuous, and an upper bound for the jumps up is given. Furthermore the cont...

On any compact Riemannian manifold (M,g) of dimension n, the Lnormalized eigenfunctions {φλ} satisfy ||φλ||∞ ≤ Cλ n−1 2 where −∆φλ = λ 2φλ. The bound is sharp in the class of all (M, g) since it is obtained by zonal spherical harmonics on the standard n-sphere S. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori R/Γ. We say that S, but not R/Γ, is a Riemannian manifol...

We compute symmetry algebras for nonlinear Schrödinger equations which contain an imaginary nonlinearity as derived by Doebner and Goldin and certain real nonlinearities not depending on the derivatives. In the three-dimensional case we find the maximal symmetry algebras for equations of this type. Admitting other imaginary nonlinearities does lead to similar symmetry algebras. These symmetries...

The problem of nonparametric regression is addressed, and a kernel smoothing estimator is proposed which has favorable asymptotic performance (bias, variance, and mean squared error). The proposed class of kernels is characterized by a Fourier transform which is flat near the origin and infinitely differentiable. This property allows the bias of the estimate to decrease at the maximal rate with...

The uniqueness theorem for the ergodic maximal operator is proved in the continuous case. Let (X, S, μ) be a finite measure space, μ(X) < ∞, (1) and let (Tt)t≥0 be an ergodic semigroup of measure-preserving transformations of (X, S, μ). As usual the map (x, t) → Ttx is assumed to be jointly measurable. For an integrable function f , f ∈ L(X), the ergodic maximal function f∗ is defined by equati...

This month’s editorials question the basis of some accepted beliefs, namely the benefit of minimising the time spent as an in-patient in hospital, and the reasons for favouring atypical antipsychotics in the treatment of schizophrenia. Capdevielle & Ritchie (pp. 164–165) debate the risks and benefits of shorter hospital stays and suggest that the shortest stays occur in countries where communit...

We discuss current approaches to the development of natural language dialogue systems, and claim that they do not sufficiently consider the unique qualities of man-machine interaction as distinct from general human discourse. We conclude that empirical studies of this unique communication situation is required for the development of user-friendly interactive systems. One way of achieving this i...