A constant-time algorithm for middle levels Gray codes

For any integer $n\geq 1$ a middle levels Gray code is a cyclic listing of all $n$-element and $(n+1)$-element subsets of $\{1,2,\ldots,2n+1\}$ such that any two consecutive subsets differ in adding or removing a single element. The question whether such a Gray code exists for any $n\geq 1$ has been the subject of intensive research during the last 30 years, and has been answered affirmatively only recently [T. M\"utze. Proof of the middle levels conjecture. Proc. London Math. Soc., 112(4):677--713, 2016]. In a follow-up paper [T. M\"utze and J. Nummenpalo. An efficient algorithm for computing a middle levels Gray code. To appear in ACM Transactions on Algorithms, 2018] this existence proof was turned into an algorithm that computes each new set in the Gray code in time $\mathcal{O}(n)$ on average. In this work we present an algorithm for computing a middle levels Gray code in optimal time and space: each new set is generated in time $\mathcal{O}(1)$ on average, and the required space is $\mathcal{O}(n)$

Evidence of Environmental Dependencies of Type Ia Supernovae from the Nearby Supernova Factory indicated by Local H{\alpha}

(Abridged) We study the host galaxy regions in close proximity to Type Ia supernovae (SNe Ia) to analyze relations between the properties of SN Ia events and environments most similar to where their progenitors formed. We focus on local H\alpha\ emission as an indicator of young environments. The Nearby Supernova Factory has obtained flux-calibrated spectral timeseries for SNe Ia using integral field spectroscopy, allowing the simultaneous measurement of the SN and its immediate vicinity. For 89 SNe Ia we measure H\alpha\ emission tracing ongoing star formation within a 1 kpc radius around each SN. This constitutes the first direct study of the local environment for a large sample of SNe Ia also having accurate luminosity, color and stretch measurements. We find that SNe Ia with local H\alpha\ emission are redder by 0.036+/-0.017 mag, and that the previously-noted correlation between stretch and host mass is entirely driven by the SNe Ia coming from passive regions. Most importantly, the mean standardized brightness for SNe Ia with local H\alpha\ emission is 0.094+/-0.031 mag fainter than for those without. This offset arises from a bimodal structure in the Hubble residuals, that also explains the previously-known host-mass bias. We combine this bimodality with the cosmic star-formation rate to predict changes with redshift in the mean SN Ia brightness and the host-mass bias. This change is confirmed using high-redshift SNe Ia from the literature. These environmental dependences point to remaining systematic errors in SNe Ia standardization. The observed brightness offset is predicted to cause a significant bias in measurements of the dark energy equation of state. Recognition of these effects offers new opportunities to improve SNe Ia as cosmological probes - e.g. SNe Ia having local H\alpha\ emission are more homogeneous, having a brightness dispersion of 0.105+/-0.012 mag.Comment: accepted for publication in Section 3. Cosmology of A&A (The official date of acceptance is 30/08/2013

A Supermassive Black Hole in an Ultracompact Dwarf Galaxy

Ultracompact dwarf galaxies (UCDs) are among the densest stellar systems in the universe. These systems have masses up to 200 million solar masses, but half light radii of just 3-50 parsecs. Dynamical mass estimates show that many UCDs are more massive than expected from their luminosity. It remains unclear whether these high dynamical mass estimates are due to the presence of supermassive black holes or result from a non-standard stellar initial mass function that causes the average stellar mass to be higher than expected. Here we present the detection of a supermassive black hole in a massive UCD. Adaptive optics kinematic data of M60-UCD1 show a central velocity dispersion peak above 100 km/s and modest rotation. Dynamical modeling of these data reveals the presence of a supermassive black hole with mass of 21 million solar masses. This is 15% of the object's total mass. The high black hole mass and mass fraction suggest that M60-UCD1 is the stripped nucleus of a galaxy. Our analysis also shows that M60-UCD1's stellar mass is consistent with its luminosity, implying many other UCDs may also host supermassive black holes. This suggests a substantial population of previously unnoticed supermassive black holes.Comment: Author's version of paper appearing in 18 September issue of Nature, available at http://dx.doi.org/10.1038/nature13762 ; 9 pages, 9 figures including methods & supplementary information section