Quantum in any dimension

Monotone Subsequences in Any Dimension

We exhibit sequences of n points in d dimensions with no long monotone subsequences, by which we mean when projected in a general direction, our sequence has no monotone subsequences of length n+d or more. Previous work proved that this function of n would lie between n and 2 n; this paper establishes that the coefficient of n is one. This resolves the question of how the Erdo s Szekeres result...

Any-Dimension Algorithms

Dynamical partitions of space in any dimension

Topologically stable cellular partitions of D-dimensional spaces are studied. A complete statistical description of the average structural properties of such partitions is given in terms of a sequence of D2 − 1 (or D−1 2 ) variables for D even (or odd). These variables are the average coordination numbers of the 2k-dimensional polytopes (2k < D) which make up the cellular structure. A procedure...

The vortex filament equation in any dimension

We present the vortex filament (or localized induction approximation) equation in any dimension. For an arbitrary n ≥ 3 the evolution of vorticity supported on vortex membranes of codimension 2 in R is described by the skew (or binormal) mean-curvature flow, which generalises to any dimension the classical binormal equation in R. This paper is a brief summary of the results in Khesin (2012) and...

Quasi-Spherical Gravitational Collapse in Any Dimension

We study the occurrence and nature of naked singularities for a dust model with non-zero cosmological constant in (n + 2)-dimensional Szekeres space-times (which possess no Killing vectors) for n ≥ 2. We find that central shell-focusing singularities may be locally naked in higher dimensions but depend sensitively on the choice of initial data. In fact, the nature of the initial density determi...