Near-optimal asymmetric binary matrix partitions

We study the asymmetric binary matrix partition problem that was recently introduced by Alon et al. (WINE 2013) to model the impact of asymmetric information on the revenue of the seller in take-it-or-leave-it sales. Instances of the problem consist of an $n \times m$ binary matrix $A$ and a probability distribution over its columns. A partition scheme $B=(B_1,...,B_n)$ consists of a partition $B_i$ for each row $i$ of $A$. The partition $B_i$ acts as a smoothing operator on row $i$ that distributes the expected value of each partition subset proportionally to all its entries. Given a scheme $B$ that induces a smooth matrix $A^B$, the partition value is the expected maximum column entry of $A^B$. The objective is to find a partition scheme such that the resulting partition value is maximized. We present a $9/10$-approximation algorithm for the case where the probability distribution is uniform and a $(1-1/e)$-approximation algorithm for non-uniform distributions, significantly improving results of Alon et al. Although our first algorithm is combinatorial (and very simple), the analysis is based on linear programming and duality arguments. In our second result we exploit a nice relation of the problem to submodular welfare maximization.Comment: 17 page

The Bright Side of Dark Matter

We show that it is not possible in the absence of dark matter to construct a four-dimensional metric that explains galactic observations. In particular, by working with an effective potential it is shown that a metric which is constructed to fit flat rotation curves in spiral galaxies leads to the wrong sign for the bending of light i.e. repulsion instead of attraction. Hence, without dark matter the motion of particles on galactic scales cannot be explained in terms of geodesic motion on a four- dimensional metric. This reveals a new bright side to dark matter: it is indispensable if we wish to retain the cherished equivalence principle.Comment: 7 pages, latex, no figures. Received an honorable mention in the 1999 Gravity research Foundation Essay Competition. Submitted to Phys. Rev. Let

Stability of disk galaxies in the modified dynamics

General analytic arguments lead us to expect that in the modified dynamics (MOND) self-gravitating disks are more stable than their like in Newtonian dynamics. We study this question numerically, using a particle-mesh code based on a multi-grid solver for the (nonlinear) MOND field equation. We start with equilibrium distribution functions for MOND disk models having a smoothly truncated, exponential surface-density profiles and a constant Toomre $Q$ parameter. We find that, indeed, disks of a given ``temperature'' are locally more stable in MOND than in Newtonian dynamics. As regards global instability to bar formation, we find that as the mean acceleration in the disk is lowered, the stability of the disk is increased as we cross from the Newtonian to the MOND regime. The degree of stability levels off deep in the MOND regime, as expected from scaling laws in MOND. For the disk model we use, this maximum degree of stability is similar to the one imparted to a Newtonian disk by a halo three times as massive at five disk scale lengths.Comment: 20 pages, Latex, 8 embedded figures, version to be published in The Astrophys.

Modified gravity without dark matter

On an empirical level, the most successful alternative to dark matter in bound gravitational systems is the modified Newtonian dynamics, or MOND, proposed by Milgrom. Here I discuss the attempts to formulate MOND as a modification of General Relativity. I begin with a summary of the phenomenological successes of MOND and then discuss the various covariant theories that have been proposed as a basis for the idea. I show why these proposals have led inevitably to a multi-field theory. I describe in some detail TeVeS, the tensor-vector-scalar theory proposed by Bekenstein, and discuss its successes and shortcomings. This lecture is primarily pedagogical and directed to those with some, but not a deep, background in General RelativityComment: 28 pages, 10 figures, lecture given at Third Aegean Summer School, The Invisible Universe: Dark Matter and Dark Energy, minor errors corrected, references update

On the Possibility of Quantum Gravity Effects at Astrophysical Scales

The nonperturbative renormalization group flow of Quantum Einstein Gravity (QEG) is reviewed. It is argued that at large distances there could be strong renormalization effects, including a scale dependence of Newton's constant, which mimic the presence of dark matter at galactic and cosmological scales.Comment: LaTeX, 18 pages, 4 figures. Invited contribution to the Int. J. Mod. Phys. D special issue on dark matter and dark energ

Cosmological extrapolation of MOND

Regime of MOND, which is used in astronomy to describe the gravitating systems of island type without the need to postulate the existence of a hypothetical dark matter, is generalized to the case of homogeneous distribution of usual matter by introducing a linear dependence of the critical acceleration on the size of region under consideration. We show that such the extrapolation of MOND in cosmology is consistent with both the observed dependence of brightness on the redshift for type Ia supernovae and the parameters of large-scale structure of Universe in the evolution, that is determined by the presence of a cosmological constant, the ordinary matter of baryons and electrons as well as the photon and neutrino radiation without any dark matter.Comment: 20 pages, 5 figures, comments adde

Phenomenological covariant approach to gravity

We covariantly modify the Einstein-Hilbert action such that the modified action perturbatively resolves the flat rotational velocity curve of the spiral galaxies and gives rise to the Tully-Fisher relation, and dynamically generates the cosmological constant. This modification requires introducing just a single new universal parameter.Comment: v6: a mistake in deriving the equation of the cosmological constant corrected, refs adde

Testing Modified Newtonian Dynamics with Rotation Curves of Dwarf and Low Surface Brightness Galaxies

Dwarf and low surface brightness galaxies are ideal objects to test modified Newtonian dynamics (MOND), because in most of these galaxies the accelerations fall below the threshold below where MOND supposedly applies. We have selected from the literature a sample of 27 dwarf and low surface brightness galaxies. MOND is successful in explaining the general shape of the observed rotation curves for roughly three quarters of the galaxies in the sample presented here. However, for the remaining quarter, MOND does not adequately explain the observed rotation curves. Considering the uncertainties in distances and inclinations for the galaxies in our sample, a small fraction of poor MOND predictions is expected and is not necessarily a problem for MOND. We have also made fits taking the MOND acceleration constant, a_0, as a free parameter in order to identify any systematic trends. We find that there appears to be a correlation between central surface brightness and the best-fit value of a_0, in the sense that lower surface brightness galaxies tend to have lower a_0. However, this correlation depends strongly on a small number of galaxies whose rotation curves might be uncertain due to either bars or warps. Without these galaxies, there is less evidence of a trend, but the average value we find for a_0 ~ 0.7*10^-8 cm s^-2 is somewhat lower than derived from previous studies. Such lower fitted values of a_0 could occur if external gravitational fields are important.Comment: 12 pages, accepted for publication in Ap

The Mass of the Compact Object in the X-Ray Binary Her X-1/HZ Her

We have obtained the first estimates of the masses of the components of the Her X-1/HZ Her X-ray binary system taking into account non-LTE effects in the formation of the H_gamma absorption line: mx=1.8Msun and mv=2.5Msun. These mass estimates were made in a Roche model based on the observed radial-velocity curve of the optical star, HZ Her. The masses for the X-ray pulsar and optical star obtained for an LTE model lie are mx=0.85\pm0.15Msun and mv=1.87\pm0.13Msun. These mass estimates for the components of Her X-1/HZ Her derived from the radial-velocity curve should be considered tentative. Further mass estimates from high-precision observations of the orbital variability of the absorption profiles in a non-LTE model for the atmosphere of the optical component should be made.Comment: 20 pages, 4 tables, 8 figure

Inclination Effects and Beaming in Black Hole X-ray Binaries

We investigate the dependence of observational properties of black hole X-ray binaries on the inclination angle i of their orbits. We find the following: (1) Transient black hole binaries show no trend in their quiescent X-ray luminosities as a function of i, suggesting that the radiation is not significantly beamed. This is consistent with emission from an accretion disk. If the X-rays are from a jet, then the Lorentz factor gamma of the jet is less than 1.24 at the 90% confidence level. (2) The X-ray binary 4U1543-47 with i of order 21 degrees has a surprisingly strong fluorescent iron line in the high soft state. Quantifying an earlier argument by Park et al. (2004), we conclude that if the continuum X-ray emission in this source is from a jet, then gamma < 1.04. (3) None of the known binaries has cos i 75 degrees. This fact, plus the lack of eclipses among the 20 black hole binaries in our sample, strongly suggests at the 99.5% confidence level that systems with large inclination angles are hidden from view. The obscuration could be the result of disk flaring, as suggested by Milgrom (1978) for neutron star X-ray binaries. (4) Transient black hole binaries with i ~ 70-75 degrees have significantly more complex X-ray light curves than systems with i < 65 degrees. This may be the result of variable obscuration and/or variable height above the disk of the radiating gas.Comment: 26 pages, to appear in The Astrophysical Journal, vol. 624, May 1, 200