Average resistance of toroidal graphs

The average effective resistance of a graph is a relevant performance index in many applications, including distributed estimation and control of network systems. In this paper, we study how the average resistance depends on the graph topology and specifically on the dimension of the graph. We concentrate on $d$-dimensional toroidal grids and we exploit the connection between resistance and Laplacian eigenvalues. Our analysis provides tight estimates of the average resistance, which are key to study its asymptotic behavior when the number of nodes grows to infinity. In dimension two, the average resistance diverges: in this case, we are able to capture its rate of growth when the sides of the grid grow at different rates. In higher dimensions, the average resistance is bounded uniformly in the number of nodes: in this case, we conjecture that its value is of order $1/d$ for large $d$. We prove this fact for hypercubes and when the side lengths go to infinity.Comment: 24 pages, 6 figures, to appear in SIAM Journal on Control and Optimization (SICON

Making sense of the local Galactic escape speed estimates in direct dark matter searches

Direct detection (DD) of dark matter (DM) candidates in the $\lesssim$10 GeV mass range is very sensitive to the tail of their velocity distribution. The important quantity is the maximum WIMP speed in the observer's rest frame, i.e. in average the sum of the local Galactic escape speed $v_{\rm esc}$ and of the circular velocity of the Sun $v_c$. While the latter has been receiving continuous attention, the former is more difficult to constrain. The RAVE Collaboration has just released a new estimate of $v_{\rm esc}$ (Piffl {\em et al.}, 2014 --- P14) that supersedes the previous one (Smith {\em et al.}, 2007), which is of interest in the perspective of reducing the astrophysical uncertainties in DD. Nevertheless, these new estimates cannot be used blindly as they rely on assumptions in the dark halo modeling which induce tight correlations between the escape speed and other local astrophysical parameters. We make a self-consistent study of the implications of the RAVE results on DD assuming isotropic DM velocity distributions, both Maxwellian and ergodic. Taking as references the experimental sensitivities currently achieved by LUX, CRESST-II, and SuperCDMS, we show that: (i) the exclusion curves associated with the best-fit points of P14 may be more constraining by up to $\sim 40$% with respect to standard limits, because the underlying astrophysical correlations induce a larger local DM density; (ii) the corresponding relative uncertainties inferred in the low WIMP mass region may be moderate, down to 10-15% below 10 GeV. We finally discuss the level of consistency of these results with other independent astrophysical constraints. This analysis is complementary to others based on rotation curves.Comment: 18 pages, 7 figures. V2: improved version that matches to the published on

Weighing the local dark matter with RAVE red clump stars

We determine the Galactic potential in the solar neigbourhood from RAVE observations. We select red clump stars for which accurate distances, radial velocities, and metallicities have been measured. Combined with data from the 2MASS and UCAC catalogues, we build a sample of 4600 red clump stars within a cylinder of 500 pc radius oriented in the direction of the South Galactic Pole, in the range of 200 pc to 2000 pc distances. We deduce the vertical force and the total mass density distribution up to 2 kpc away from the Galactic plane by fitting a distribution function depending explicitly on three isolating integrals of the motion in a separable potential locally representing the Galactic one with four free parameters. Because of the deep extension of our sample, we can determine nearly independently the dark matter mass density and the baryonic disc surface mass density. We find (i) at 1kpc Kz/(2piG) = 68.5 pm 1.0 Msun/pc2, and (ii) at 2 kpc Kz/(2piG) = 96.9 pm 2.2 Msun/pc2. Assuming the solar Galactic radius at R0 = 8.5 kpc, we deduce the local dark matter density rhoDM (z=0) = 0.0143 pm 0.0011Msun pc3 = 0.542 pm 0.042 Gev/cm3 and the baryonic surface mass density Sigma = 44.4 pm 4.1 Msun/pc2 . Our results are in agreement with previously published Kz determinations up to 1 kpc, while the extension to 2 kpc shows some evidence for an unexpectedly large amount of dark matter. A flattening of the dark halo of order 0.8 can produce such a high local density in combination with a circular velocity of 240 km/s . Another explanation, allowing for a lower circular velocity, could be the presence of a secondary dark component, a very thick disc resulting either from the deposit of dark matter from the accretion of multiple small dwarf galaxies, or from the presence of an effective phantom thick disc in the context of effective galactic-scale modifications of gravity.Comment: 14 pages, 13 figures, accepted to Astronomy and Astrophysic

These are the echoes: Sound Proof 2008-2012

Article in peer-reviewed journal Culture/Kultura for their thematic issue, Art Media and Cultural Memory. Based on the conference talk given at CCCS Annual Conference on Cultural Memory 4-7 September 2013