In the present short communication, we revisit nonlinearity management of the timeperiodic nonlinear Schrödinger equation and the related averaging procedure. By means of rigorous estimates, we show that the averaged nonlinear Schrödinger equation does not blow up in the higher dimensional case so long as the corresponding solution remains smooth. In particular, we show that the H norm remains ...

In this article we look at the well-studied upper bounds for |A|, where A ⊂ N is a Bh sequence, and generalise these to the case where A ⊂ N d. In particular we give d-dimensional analogues to results of Chen, Jia, Graham and Green.

We show that in dimensions d 3, aperiodic tilings can naturally avoid more symmetries than just translations.

A Ducci sequence is a sequence of n-tuples of integers obtained by iterating the map (a1, . . . , an) !→ (|a1 − a2|, |a2 − a3|, . . . , |an − a1|). In this paper, we generalize the concept of Ducci sequences to sequences of d-dimensional arrays, extend some of the basic results on Ducci sequences to this case, and point out some new phenomena that occur. Our main tool is the description of Ducc...

In this paper we are concerned with knight’s tours on high-dimensional boards. Our main aim is to show that on the d-dimensional board [n], with n even, there is always a knight’s tour provided that n is sufficiently large. In fact, we give an exact classification of the grids [n1] × . . . × [nd] in which there is a knight’s tour. This answers questions of DeMaio, DeMaio and Mathew, and Watkins.

We generalize the array orthogonality property for perfect autocorrelation sequences to n-dimensional arrays. The generalized array orthogonality property is used to derive a number of ndimensional perfect array constructions.

We review black-hole solutions of higher-dimensional vacuum gravity and higher-dimensional supergravity theories. The discussion of vacuum gravity is pedagogical, with detailed reviews of Myers-Perry solutions, black rings, and solution-generating techniques. We discuss black-hole solutions of maximal supergravity theories, including black holes in anti-de Sitter space. General results and open...

We explore a classical instability of spacetimes of dimension D > 4. Firstly, we consider static solutions: generalised black holes and brane world metrics. The dangerous mode is a tensor mode on an Einstein base manifold of dimension D−2. A criterion for instability is found for the generalised Schwarzschild, AdSSchwarzschild and topological black hole spacetimes in terms of the Lichnerowicz s...