Using the kinematic Sunyaev-Zeldovich effect to determine the peculiar velocities of clusters of galaxies

We have investigated the possibility of inferring peculiar velocities for clusters of galaxies from the Doppler shift of scattered cosmic microwave background (CMB) photons. We find that if the core radius of the gas distribution or the beam size of the instrument is larger than 3-7 arcminutes, then the maximum attainable signal-to-noise ratio is determined by confusion with primary fluctuations. For smaller angular scales, ``cosmic confusion'' is less important and instrumental noise and/or foreground emission will be the limiting factor. For a cluster with the optical depth of the Coma cluster and for an optimal filtering technique, typical one-sigma errors span the wide range from 400 to 1600 km/s, depending on the cosmological model, the resolution of the instrument and the core radius of the cluster. The results have important implications for the design of future high-resolution surveys of the CMB. Individual peculiar velocities will be measurable only for a few fast moving clusters at intermediate redshift unless cosmic fluctuations are smaller than most standard cosmological scenarios predict. However, a reliable measurement of bulk velocities of ensembles of X-ray bright clusters will be possible on very large scales (100-500 Mpc/h).Comment: 34 pages, with 11 figures included. Postscript. Submitted to MNRAS. Latest version (recommended) at http://www.mpa-garching.mpg.de/~max/sz.html or from [email protected]

Nagata compactification for algebraic spaces

We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian approximation for such algebraic spaces, generalizing earlier results in the case of schemes.Comment: 49 pages, various clarifications and bugfixe

Complexity and Approximation of the Continuous Network Design Problem

We revisit a classical problem in transportation, known as the continuous (bilevel) network design problem, CNDP for short. We are given a graph for which the latency of each edge depends on the ratio of the edge flow and the capacity installed. The goal is to find an optimal investment in edge capacities so as to minimize the sum of the routing cost of the induced Wardrop equilibrium and the investment cost. While this problem is considered as challenging in the literature, its complexity status was still unknown. We close this gap showing that CNDP is strongly NP-complete and APX-hard, both on directed and undirected networks and even for instances with affine latencies. As for the approximation of the problem, we first provide a detailed analysis for a heuristic studied by Marcotte for the special case of monomial latency functions (Mathematical Programming, Vol.~34, 1986). Specifically, we derive a closed form expression of its approximation guarantee for arbitrary sets S of allowed latency functions. Second, we propose a different approximation algorithm and show that it has the same approximation guarantee. As our final -- and arguably most interesting -- result regarding approximation, we show that using the better of the two approximation algorithms results in a strictly improved approximation guarantee for which we give a closed form expression. For affine latencies, e.g., this algorithm achieves a 1.195-approximation which improves on the 5/4 that has been shown before by Marcotte. We finally discuss the case of hard budget constraints on the capacity investment.Comment: 27 page