Asymptotic dimension and uniform embeddings

We show that the type function of a space with finite asymptotic dimension estimates its Hilbert (or any $l^p$) compression. The method allows to obtain the lower bound of the compression of the lamplighter group $Z\wr Z$, which has infinite asymptotic dimension.Comment: 17 pages, no figure

Euler characteristic of the configuration space of a complex

A closed form formula (generating function) for the Euler characteristic of the configuration space of $\scriptstyle n$ particles in a simplicial complex is given.Comment: 6 pages, no figure

The Universal Aspect Ratio of Vortices in Rotating Stratified Flows: Theory and Simulation

We derive a relationship for the vortex aspect ratio $\alpha$ (vertical half-thickness over horizontal length scale) for steady and slowly evolving vortices in rotating stratified fluids, as a function of the Brunt-Vaisala frequencies within the vortex $N_c$ and in the background fluid outside the vortex $\bar{N}$, the Coriolis parameter $f$, and the Rossby number $Ro$ of the vortex: $\alpha^2 = Ro(1+Ro) f^2/(N_c^2-\bar{N}^2)$. This relation is valid for cyclones and anticyclones in either the cyclostrophic or geostrophic regimes; it works with vortices in Boussinesq fluids or ideal gases, and the background density gradient need not be uniform. Our relation for $\alpha$ has many consequences for equilibrium vortices in rotating stratified flows. For example, cyclones must have $N_c^2 > \bar{N}^2$; weak anticyclones (with $|Ro| \bar{N}^2$. We verify our relation for $\alpha$ with numerical simulations of the three-dimensional Boussinesq equations for a wide variety of vortices, including: vortices that are initially in (dissipationless) equilibrium and then evolve due to an imposed weak viscous dissipation or density radiation; anticyclones created by the geostrophic adjustment of a patch of locally mixed density; cyclones created by fluid suction from a small localised region; vortices created from the remnants of the violent breakups of columnar vortices; and weakly non-axisymmetric vortices. The values of the aspect ratios of our numerically-computed vortices validate our relationship for $\alpha$, and generally they differ significantly from the values obtained from the much-cited conjecture that $\alpha = f/\bar{N}$ in quasi-geostrophic vortices.Comment: Submitted to the Journal of Fluid Mechanics. Also see the companion paper by Aubert et al. "The Universal Aspect Ratio of Vortices in Rotating Stratified Flows: Experiments and Observations" 201

Photometric validation of a model independent procedure to extract galaxy clusters

By means of CCD photometry in three bands (Gunn g, r, i) we investigate the existence of 12 candidate clusters extracted via a model independent peak finding algorithm (\cite{memsait}) from DPOSS data. The derived color-magnitude diagrams allow us to confirm the physical nature of 9 of the cluster candidates, and to estimate their photometric redshifts. Of the other candidates, one is a fortuitous detection of a true cluster at z~0.4, one is a false detection and the last is undecidable on the basis of the available data. The accuracy of the photometric redshifts is tested on an additional sample of 8 clusters with known spectroscopic redshifts. Photometric redshifts turn out to be accurate within z~0.01 (interquartile range).Comment: A&A in pres

The Universal Aspect Ratio of Vortices in Rotating Stratifi?ed Flows: Experiments and Observations

We validate a new law for the aspect ratio $\alpha = H/L$ of vortices in a rotating, stratified flow, where $H$ and $L$ are the vertical half-height and horizontal length scale of the vortices. The aspect ratio depends not only on the Coriolis parameter f and buoyancy (or Brunt-Vaisala) frequency $\bar{N}$ of the background flow, but also on the buoyancy frequency $N_c$ within the vortex and on the Rossby number $Ro$ of the vortex such that $\alpha = f \sqrt{[Ro (1 + Ro)/(N_c^2- \bar{N}^2)]}$. This law for $\alpha$ is obeyed precisely by the exact equilibrium solution of the inviscid Boussinesq equations that we show to be a useful model of our laboratory vortices. The law is valid for both cyclones and anticyclones. Our anticyclones are generated by injecting fluid into a rotating tank filled with linearly-stratified salt water. The vortices are far from the top and bottom boundaries of the tank, so there is no Ekman circulation. In one set of experiments, the vortices viscously decay, but as they do, they continue to obey our law for $\alpha$, which decreases over time. In a second set of experiments, the vortices are sustained by a slow continuous injection after they form, so they evolve more slowly and have larger |Ro|, but they also obey our law for $\alpha$. The law for $\alpha$ is not only validated by our experiments, but is also shown to be consistent with observations of the aspect ratios of Atlantic meddies and Jupiter's Great Red Spot and Oval BA. The relationship for $\alpha$ is derived and examined numerically in a companion paper by Hassanzadeh et al. (2012).Comment: Submitted to the Journal of Fluid Mechanics. Also see the companion paper by Hassanzadeh et al. "The Universal Aspect Ratio of Vortices in Rotating Stratifi?ed Flows: Theory and Simulation" 201