It is shown that if (vj)n1 is a sequence of norm 1 vectors in a complex Hilbert space and (tj)n1 is a sequence of non-negative numbers satisfying

[formula here]

then there is a unit vector z for which

[formula here]

for every j. The result is a strong, complex analogue of the author's real plank theorem.

The purpose of this paper is to give sufficient conditions for all nontrivial solutions of the nonlinear differential equation x″ +a(t)g(x) = 0 to be nonoscillatory. Here, g(x) satisfies the sign condition xg(x) > 0 if x ≠ 0, but is not assumed to be monotone increasing. This differential equation includes the generalized Emden–Fowler equation as a special case. Our main result extends some nonoscillation theorems for the generalized Emden–Fowler equation. Proof is given by means of some Liapunov functions and phase-plane analysis.

The dual of a reflexive separable hereditarily indecomposable complex Banach space of Gowers and Maurey admits no chaotic continuous linear operator in the sense of Devaney defined on it.

We prove that for an arbitrary measurable set A ⊂ ℝ2 and a σ-finite Borel measure μ on the plane, there is a Borel set of lines L such that for each point in A, the set of directions of those lines from L containing the point is a residual set, and, moreover, μ(A) = μ({∪[lscr ] : [lscr ] ∈ L}). We show how this result may be used to characterise the sets of the plane from which an invisible set is visible. We also characterise the rectifiable sets C1, C2 for which there is a set which is visible from C1 and invisible from C2.

Let X be an infinite-dimensional Banach space, and let A be a Cp Lipschitz bounded starlike body (for instance the unit ball of a smooth norm). We prove that:

(1) the boundary ∂A is Cp Lipschitz contractible;

(2) there is a Cp Lipschitz retraction from A onto ∂A;

(3) there is a Cp Lipschitz map T : A → A with no approximate fixed points.

Let f be a holomorphic function on the strip {z ∈ [Copf ] : −α < Im z < α}, where α > 0, belonging to the class [Hscr ](α,−α;ε) defined below. It is shown that there exist holomorphic functions w1 on {z ∈ [Copf ] : 0 < Im z < 2α} and w2 on {z ∈ [Copf ] : −2α < Im z < 2α}, such that w1 and w2 have boundary values of modulus one on the real axis, and satisfy the relations

w1(z)=f(z-αi)w2(z-2αi) and w2(z+2αi)=f(z+αi)w1(z)

for 0 < Im z < 2α, where f(z) := f(z). This leads to a ‘polar decomposition’ f(z) = uf(z + αi)gf(z) of the function f(z), where uf (z + αi) and gf(z) are holomorphic functions for −α < Im z < α, such that [mid ]uf(x)[mid ] = 1 and gf(x) [ges ] 0 almost everywhere on the real axis. As a byproduct, an operator representation of a q-deformed Heisenberg algebra is developed.

It is known that, if f is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set, J(f), are equal. In this paper it is shown that, for a hyperbolic transcendental meromorphic function f, the packing and upper box dimensions of J(f) are equal, but can be strictly greater than the Hausdorff dimension of J(f).

This paper introduces a stratification of the variety of modules of a fixed dimension over a finite-dimensional algebra. For a large and interesting class of algebras, called subfinite, this stratification is used to compute the dimension of this variety and give bounds on the number of its irreducible components.

Recall that if S is a class of groups, then a group G is residually-S if, for any element 1 ≠ g ∈ G, there is a normal subgroup N of G such that g ∉ N and G/N ∈ S. Let Λ be a commutative Noetherian local pro-p ring, with a maximal ideal M. Recall that the first congruence subgroup of SLd(Λ) is: SL1d(Λ) = ker (SLd(Λ) → SLd(Λ/M)).

Let K ⊆ ℕ. We define SΛ(K) = ∪d∈K{open subgroups of SL1d(Λ)}. We show that if K is infinite, then for Λ = [ ]p[[t]] and for Λ = ℤp a finitely generated non-abelian free pro-p group is residually-SΛ(K). We apply a probabilistic method, combined with Lie methods and a result on random generation in simple algebraic groups over local fields. It is surprising that the case of zero characteristic is deduced from the positive characteristic case.

Conditions are given which imply that analytic iterated function systems (IFSs) in the complex plane [Copf ] have uniformly perfect attractor sets. In particular, it is shown that the attractor set of a finitely generated conformal IFS is uniformly perfect when it contains two or more points. Also, an example is given of a finitely generated analytic attractor set which is not uniformly perfect.

In this paper, a new approach to the Calabi–Bernstein theorem on maximal surfaces in the Lorentz– Minkowski space L3 is introduced. The approach is based on an upper bound for the total curvature of geodesic discs in a maximal surface in L3, involving the local geometry of the surface and its hyperbolic image. As an application of this, a new proof of the Calabi–Bernstein theorem is provided.

Let f be analytic in the unit disc, and let it belong to the Hardy space Hp, equipped with the usual norm ∥f∥p. It is known from the work of Hardy and Littlewood that for q > p, the constants

[formula here]

with the usual extension to the case where q = ∞, have C(p, q) < ∞ .The authors prove that

[formula here]

and

[formula here]

We prove uniqueness for the nonharmonic trigonometric series [sum ]∞k=0akeiλkx under the weaker condition (*) where [sum ]∞k=0 [mid ]ak[mid ]/exp[mid ]λk[mid ]γ < ∞, for some 0 < γ < 1. In other words, if {λk}∞0 satisfies the above condition (*), and if [sum ]∞k=0akeiλkx = 0, then ak = 0 for all k = 0, 1,…. Finally, we state an improvement of Zygmund's uniqueness result as a corollary.

It is shown that if a C*-algebra A contains a semi-scattered C*-algebra B such that the pure states of B and the zero functional extend uniquely to A, and the canonical mapping Bˆ → Â is injective, then there exists a (unique) projection of norm one R : A → B. In certain circumstances, the conditional expectation R can be effected by a unitary averaging process using unitary elements in the centre of the multiplier algebra M(B).

If we consider the 14-sided hyperbolic polygon of Felix Klein that defines his famous surface of genus 3, then we observe that we have a uniform, unifacial dessin whose automorphism group is transitive on the edges, but not on the directed edges of the dessin. We show that Klein's surface is the unique platonic surface with this property.

Let M be a simply connected closed manifold of dimension greater than 4 which does not admit a metric with positive scalar curvature. We give necessary conditions for M to admit a scalar-flat metric. These conditions involve the first Pontrjagin class and the cohomology ring of M. As a consequence, any simply connected scalar-flat manifold of dimension greater than 4 with vanishing first Pontrjagin class admits a metric with positive scalar curvature. We also describe some relations between scalar-flat metrics, almost complex structures and the free loop space.

Let (G, X) be a locally compact transformation group in which G acts freely on X. We show that the associated transformation-group C*-algebra C0(X) [rtimes ] G is a Fell algebra if and only if X is a Cartan G-space.

First, we shall prove that a compact connected oriented locally conformally flat n-dimensional Riemannian manifold with constant scalar curvature is isometric to a space form or a Riemannian product Sn−1(c) × S1 if its Ricci curvature is nonnegative. Second, we shall give a topological classification of compact connected oriented locally conformally flat n-dimensional Riemannian manifolds with nonnegative scalar curvature r if the following inequality is satisfied: [sum ]i,jR2ij [les ] r2/(n−1), where [sum ]i,jR2ij is the squared norm of the Ricci curvature tensor.

For every infinite cardinal κ, we construct a primitive permutation group which has a finite suborbit paired with a suborbit of size κ. This answers a question of Peter M. Neumann.

In this paper, we prove the existence of an unbounded sequence of weak solutions under an appropriate oscillating behaviour of φ for a Neumann problem of the type