Observed Non-Steady State Cooling and the Moderate Cluster Cooling Flow Model

We examine recent developments in the cluster cooling flow scenario following recent observations by Chandra and XMM-Newton. We show that the distribution of gas emissivity verses temperature determined by XMM-Newton gratings observations demonstrates that the central gas in cooling flow clusters cannot be in simple steady-state. Combining this result with the lack of spectroscopic evidence for gas below one-third of the ambient cluster temperature is strong evidence that the gas is heated intermittently. While the old steady-state isobaric cooling flow model is incompatible with recent observations, a "moderate cooling flow model", in which the gas undergoes intermittent heating that effectively reduces the age of a cooling flow is consistent with observations. Most of the gas within cooling flows resides in the hottest gas, which is prevented from cooling continuously and attaining a steady-state configuration. This results in a mass cooling rate that decreases with decreasing temperature, with a much lower mass cooling rate at the lowest temperatures. The present paper strengthens the moderate cooling flow model, which can accommodate the unique activities observed in cooling flow clusters.Comment: ApJ, in pres

A parabolic Harnack principle for balanced difference equations in random environments

We consider difference equations in balanced, i.i.d. environments which are not necessary elliptic. In this setting we prove a parabolic Harnack inequality (PHI) for non-negative solutions to the discrete heat equation satisfying a (rather mild) growth condition, and we identify the optimal Harnack constant for the PHI. We show by way of an example that a growth condition is necessary and that our growth condition is sharp. Along the way we also prove a parabolic oscillation inequality and a (weak) quantitative homogenization result, which we believe to be of independent interest.Comment: 35 pages, 3 figures ; Some references where updated compared to previous versio

Inapproximability of Truthful Mechanisms via Generalizations of the VC Dimension

Algorithmic mechanism design (AMD) studies the delicate interplay between computational efficiency, truthfulness, and optimality. We focus on AMD's paradigmatic problem: combinatorial auctions. We present a new generalization of the VC dimension to multivalued collections of functions, which encompasses the classical VC dimension, Natarajan dimension, and Steele dimension. We present a corresponding generalization of the Sauer-Shelah Lemma and harness this VC machinery to establish inapproximability results for deterministic truthful mechanisms. Our results essentially unify all inapproximability results for deterministic truthful mechanisms for combinatorial auctions to date and establish new separation gaps between truthful and non-truthful algorithms