Suppose that an ordinary differential equation in ℝ4 has an orbit Γ bi-asymptotic to a stationary point O of the flow. If the characteristic equation of the linear flow near O has roots (λ0 ± ιω0, λ1±ιω1) with —λ0 > λ1 > 0 and ωi > 0 for i = 0,1, we show that there are sequences of more complicated orbits bi-asymptotic to O in generic one-parameter perturbations of the equations. When ω0 ≠ 2nω1 for all n ∈ ℕ, there are sequences of such orbits on both sides of the bifurcation value of the parameter, in contrast to a similar case for flows in ℝ3.

The following problem is the simplest of a group of problems motivated by work of Durran and Truman [1] on planetesimals.

Let {Xi. i ≥1} be a sequence of independent exponential random variables with mean 1. Setting S(n) = X1 +…+ Xn, S(0) = 0, we define

where f is a real-valued, measurable function satisfying

.

In [1] Chan and Williams considered a one-dimensional diffusion of the form

where F is a strictly increasing continuous function with F(0) = 0 and ε is a decreasing deterministic function such that ε(0) is finite and ε(t) ↓ 0 as t↑ ∞, and gave necessary and sufficient conditions for Yt →0 a.s. as t→∞.

A generalization of the classical theorems of Paley and Wiener[5] and Plancherel and Polya[6] concerning entire functions of exponential type is obtained. The proof relies only on the Cauchy theorem and the Hardy–Littlewood inequality for the Fourier transform (see [8, 9]). Since the functions under consideration are supposed to be defined only in two opposite octants in ℂn, a version of the edge of the wedge theorem [7] is derived as a by-product.

KMS states of a twisted convolution algebra of Schwartz functions on a vector group are classified and related to KMS states of twisted L1-algebras for certain subquotients. The KMS states for the subquotient algebras are also related to Fock states of vector groups. In the particular case of the subquotient Tn × ℤn of ℚ2n this links the Fock space construction of the theta functions with their appearance in KMS states of loop groups and in the Kac character formula.

In this paper we derive an expression for the point source Green's function for the reduced wave equation, valid in an angular sector whose angle is equal to a rational multiple of π. This Green's function can be used to find new expressions for the field produced by the diffraction of a spherical wave by a wedge whose angle can be expressed as a rational multiple of π. The expressions obtained will be in the form of source terms and real integrals representing the diffracted field. The general result obtained is used to derive a new representation for the solution of the problem of diffraction by a mixed hard–soft half plane.

A non-singular solution of the bosonic SU(1, 1)/U(1) no-scale supergravity field equations is constructed for an anisotropic cosmological model which is of Bianchi type VIIh.