Self-Organizing Maps and the US Urban Spatial Structure

This article considers urban spatial structure in US cities using a multi- dimensional approach. We select six key variables (commuting costs, den- sity, employment dispersion/concentration, land-use mix, polycentricity and size) from the urban literature and define measures to quantify them. We then apply these measures to 359 metropolitan areas from the 2000 US Census. The adopted methodological strategy combines two novel techniques for the social sciences to explore the existence of relevant pat- terns in such multi-dimensional datasets. Geodesic self-organizing maps (SOM) are used to visualize the whole set of information in a meaningful way, while the recently developed clustering algorithm of the max-p is applied to draw boundaries within the SOM and analyze which cities fall into each of them. JEL C45, R0, R12, R14. Keywords Urban spatial structure, self-organizing maps, US metropolitan areas

Why does gravitational radiation produce vorticity?

We calculate the vorticity of world--lines of observers at rest in a Bondi--Sachs frame, produced by gravitational radiation, in a general Sachs metric. We claim that such an effect is related to the super--Poynting vector, in a similar way as the existence of the electromagnetic Poynting vector is related to the vorticity in stationary electrovacum spacetimes.Comment: 9 pages; to appear in Classical and Quantum Gravit

Electromagnetic radiation produces frame dragging

It is shown that for a generic electrovacuum spacetime, electromagnetic radiation produces vorticity of worldlines of observers in a Bondi--Sachs frame. Such an effect (and the ensuing gyroscope precession with respect to the lattice) which is a reminiscence of generation of vorticity by gravitational radiation, may be linked to the nonvanishing of components of the Poynting and the super--Poynting vectors on the planes othogonal to the vorticity vector. The possible observational relevance of such an effect is commented.Comment: 8 pages RevTex 4-1; updated version to appear in Physical Review

Ergodicity Breaking in a Deterministic Dynamical System

The concept of weak ergodicity breaking is defined and studied in the context of deterministic dynamics. We show that weak ergodicity breaking describes a weakly chaotic dynamical system: a nonlinear map which generates subdiffusion deterministically. In the non-ergodic phase non-trivial distribution of the fraction of occupation times is obtained. The visitation fraction remains uniform even in the non-ergodic phase. In this sense the non-ergodicity is quantified, leading to a statistical mechanical description of the system even though it is not ergodic.Comment: 11 pages, 4 figure

Conserved Matter Superenergy Currents for Orthogonally Transitive Abelian G2 Isometry Groups

In a previous paper we showed that the electromagnetic superenergy tensor, the Chevreton tensor, gives rise to a conserved current when there is a hypersurface orthogonal Killing vector present. In addition, the current is proportional to the Killing vector. The aim of this paper is to extend this result to the case when we have a two-parameter Abelian isometry group that acts orthogonally transitive on non-null surfaces. It is shown that for four-dimensional Einstein-Maxwell theory with a source-free electromagnetic field, the corresponding superenergy currents lie in the orbits of the group and are conserved. A similar result is also shown to hold for the trace of the Chevreton tensor and for the Bach tensor, and also in Einstein-Klein-Gordon theory for the superenergy of the scalar field. This links up well with the fact that the Bel tensor has these properties and the possibility of constructing conserved mixed currents between the gravitational field and the matter fields.Comment: 15 page

Derivation of Source-Free Maxwell and Gravitational Radiation Equations by Group Theoretical Methods

We derive source-free Maxwell-like equations in flat spacetime for any helicity "j" by comparing the transformation properties of the 2(2j+1) states that carry the manifestly covariant representations of the inhomogeneous Lorentz group with the transformation properties of the two helicity "j" states that carry the irreducible representations of this group. The set of constraints so derived involves a pair of curl equations and a pair of divergence equations. These reduce to the free-field Maxwell equations for j=1 and the analogous equations coupling the gravito-electric and the gravito-magnetic fields for j=2.Comment: 15 pages, no figures, to appear in Int. J. Mod. Phys.