James A. Vickers

The use of generalized functions and distributions in general relativity

An uninterruptible power supply (UPS) provides improved reliability by extending the useful life of a bank of batteries that provide backup power to the UPS in the event of an extended power outage. In the preferred embodiment, battery life is extended by switchably isolating the bank of batteries from the DC buss that is coupled to the critical load. Backup power for short duration outages, on the order of about ten seconds or less, is provided by a flywheel energy storage unit. Once the outage becomes extended, the isolation circuit is triggered to electrically connect the bank of batteries directly to the DC buss. In this manner, the bank of batteries does not experience AC ripple or the common, short duration outages that would otherwise cause the batteries to experience a discharge/recharge cycle.

Diophantine approximation and a lower bound for Hausdorff dimension

Sets of the general form where U is a subset of ℛ k and is a family of subsets of U indexed by a set J , are common in the theory of Diophantine approximation [4, 7, 18, 19]. They are also closely connected with exceptional sets arising in analysis and with sets of “small divisors” in dynamical systems [1, 8, 15”. When J is the set of positive integers ℕ, the set Λ(ℱ) is of course the lim-sup of the sequence of sets F j , j = 1, 2,… [11, p. 1]. We will also call sets of the form (1), with the more general index set J, lim-sup sets. When such lim-sup sets have Lebesgue measure zero, it is of interest to determine their Hausdorff dimension. It is usually difficult to obtain a good lower bound for the Hausdorff dimension (and it can be much harder to determine than an upper bound). In this paper we will obtain a lower bound for the dimension of lim-sup sets of the form (1) for a fairly general class of families ℕ which includes a range of results in the theory of Diophantine approximation. This lower bound depends explicitly on the geometric structure and distribution in U of the sets F α in ℕ.

Number theory and dynamical systems

1. Non-degeneracy in the perturbation theory of integrable dynamical systems Helmut Riissmann 2. Infinite dimensional inverse function theorems and small divisors J. A. G. Vickers 3. Metric Diophantine approximation of quadratic forms S. J. Patterson 4. Symbolic dynamics and Diophantine equations Caroline Series 5. On badly approximable numbers, Schmidt games and bounded orbits of flows S. G. Dani 6. Estimates for Fourier coefficients of cusp forms S. Raghavan and R. Weissauer 7. The integral geometry of fractals K. J. Falconer 8. Geometry of algebraic continued fractals J. Harrison 9. Chaos implies confusion Michel Mendes France 10. The Riemann hypothesis and the Hamiltonian of a quantum mechanical system J. V. Armitage.

Generalized hyperbolicity in conical spacetimes

Solutions of the wave equation in a spacetime containing a thin cosmic string are examined in the context of nonlinear generalized functions. Existence and uniqueness of solutions to the wave equation in the Colombeau algebra is established for a conical spacetime and this solution is shown to be associated with a distributional solution. A concept of generalized hyperbolicity, based on test fields, can be defined for such singular spacetimes and it is shown that a conical spacetime is -hyperbolic.

A simple proof of the positivity of the Bondi mass

Using a nonsingular hypersurface that is not only asymptotically null, the authors prove the existence of spinor fields that yield a positive expression for the Bondi mass. Standard existence theorems based on compact hypersurfaces are applied.

Intrinsic Characterization of Manifold-Valued Generalized Functions

The concept of generalized functions taking values in a differentiable manifold is extended to a functorial theory. We establish several characterization results which allow a global intrinsic formulation both of the theory of manifold-valued generalized functions and of generalized vector bundle homomorphisms. As a consequence, a characterization of equivalence that does not resort to derivatives (analogous to scalar-valued cases of Colombeaus construction) is achieved. These results are employed to derive a point value description of all types of generalized functions valued in manifolds and to show that composition can be carried out unrestrictedly. Finally, a new concept of association adapted to the present setting is introduced.

Metric Diophantine approximation and Hausdorff dimension on manifolds

In this paper we discuss homogeneous Diophantine approximation of points on smooth manifolds M in ℝ k . We begin with a brief survey of the notation and results. For any x,y ∈ℝ k , let .

Generalised connections and curvature

The concept of generalised (in the sense of Colombeau) connection on a principal fibre bundle is introduced. This definition is then used to extend results concerning the geometry of principal fibre bundles to those that only have a generalised connection. Some applications to singular solutions of Yang–Mills theory are given.

An inequality relating total mass and the area of a trapped surface in general relativity

Let M be a space-time which is asymptotically flat at past null infinity I-, satisfies the dominant energy condition and contains a trapped surface T. The authors show that if T can be connected to I- by means of a nonsingular null-hypersurface N, then m2>or=A/16 pi where m is the Bondi mass with respect to N and A in the area of T.

Generalized flows and singular ODEs on differentiable manifolds

Based on the concept of manifold-valued generalized functions, we initiate a study of nonlinear ordinary differential equations with singular (in particular: distributional) right-hand sides in a global setting. After establishing several existence and uniqueness results for solutions of such equations and flows of singular vector fields, we compare the solution concept employed here with the purely distributional setting. Finally, we derive criteria securing that a sequence of smooth flows corresponding to the regularization of a given singular vector field converges to a measurable limiting flow.