The Fourier transform is often used to connect the Lorentzian energy distribution for resonance scattering to the exponential time dependence for decaying states. However, to apply the Fourier transform, one has to bend the rules of standard quantum mechanics; the Lorentzian energy distribution must be extended to the full real axis −∞ < E < ∞ instead of being bounded from below 0 ≤ E < ∞ (“Fer...

This paper develops a supervised dimensionality reduction method, Lorentzian Discriminant Projection (LDP), for discriminant analysis and classification. Our method represents the structures of sample data by a manifold, which is furnished with a Lorentzian metric tensor. Different from classic discriminant analysis techniques, LDP uses distances from points to their within-class neighbors and ...

Introduction Lineshape distortions due to residual eddy currents and magnetic field inhomogeneities are often present in short echo-time H spectroscopic data. If left uncorrected, these lineshape distortions lead to errors in metabolite concentration estimates when using quantification methods that incorporate model functions with specific lineshapes (i.e., Lorentzian or Gaussian). Several meth...

We study classically and quantum mechanically the Euclidean geometries compatible with an open inflationary universe of a Lorentzian geometry. The Lorentzian geometry of the open universe with an ordinary matter state matches either an open or a closed Euclidean geometry at the cosmological singularity. With an exotic matter state it matches only the open Euclidean geometry and describes a genu...

The Leech lattice has many strange properties, discovered by Conway, Parker, and Sloane. For example, it has covering radius √ 2, and the orbits of points at distance at least √ 2 from all lattice points correspond to the Niemeier lattices other than the Leech lattice. (See Conway and Sloane [6, Chaps. 22-28].) Most of the properties of the Leech lattice follow from the fact that it is the Dynk...

‎We consider four-dimensional lie groups equipped with‎ ‎left-invariant Lorentzian Einstein metrics‎, ‎and determine the harmonicity properties ‎of vector fields on these spaces‎. ‎In some cases‎, ‎all these vector fields are critical points for the energy functional ‎restricted to vector fields‎. ‎We also classify vector fields defining harmonic maps‎, ‎and calculate explicitly the energy of t...

We consider the automorphism groups of various Lorentzian lattices over the Eisenstein, Gaussian, and Hurwitz integers, and in some of them we nd reeection groups of nite index. These provide explicit constructions of new nite-covolume reeection groups acting on complex and quaternionic hyperbolic spaces of high dimensions. Speciically, we provide groups acting on C H n for all n < 6 and n = 7,...

In this paper we present an existence and multiplicity result for periodic trajectories on stationary Lorentzian manifolds, possibly with boundary, whose proof is based on a Morse theory approach, see [5]. We recall that a Lorentzian manifold is a smooth connected nite-dimensional manifold M equipped with a (0; 2) tensor eld g such that for any z ∈ M g(z) [·; ·] is a nondegenerate symmetric bil...