Tidal Tail Ejection as a Signature of Type~Ia Supernovae from White Dwarf Mergers

The merger of two white dwarfs may be preceded by the ejection of some mass in "tidal tails", creating a circumstellar medium around the system. We consider the variety of observational signatures from this material, which depend on the lag time between the start of the merger and the ultimate explosion (assuming one occurs) of the system in a Type Ia supernova. If the time lag is fairly short, the interaction of the supernova ejecta with the tails could lead to detectable shock emission at radio, optical, and/or x-ray wavelengths. At somewhat later times, the tails produce relatively broad NaID absorption lines with velocity widths of order the white dwarf escape speed ($\sim 1000$ \kms). That none of these signatures have been detected in normal SNe Ia constrains the lag time to be either very short ($\lesssim 100$ s) or fairly long ($\gtrsim 100$ yr). If the tails have expanded and cooled over timescales $\sim 10^4$ yr, they could be observable through narrow NaID and CaII H&K absorption lines in the spectra, which are seen in some fraction of SNe Ia. Using a combination of 3D and 1D hydrodynamical codes, we model the mass-loss from tidal interactions in binary systems, and the subsequent interactions with the interstellar medium, which produce a slow-moving, dense shell of gas. We synthesize NaID line profiles by ray-casting through this shell, and show that in some circumstances tidal tails could be responsible for narrow absorptions similar to those observed.Comment: 8 pages, 5 figures. Under review at Ap

A linear lower bound for incrementing a space-optimal integer representation in the bit-probe model

We present the first linear lower bound for the number of bits required to be accessed in the worst case to increment an integer in an arbitrary space- optimal binary representation. The best previously known lower bound was logarithmic. It is known that a logarithmic number of read bits in the worst case is enough to increment some of the integer representations that use one bit of redundancy, therefore we show an exponential gap between space-optimal and redundant counters. Our proof is based on considering the increment procedure for a space optimal counter as a permutation and calculating its parity. For every space optimal counter, the permutation must be odd, and implementing an odd permutation requires reading at least half the bits in the worst case. The combination of these two observations explains why the worst-case space-optimal problem is substantially different from both average-case approach with constant expected number of reads and almost space optimal representations with logarithmic number of reads in the worst case.Comment: 12 pages, 4 figure

Expected Window Mean-Payoff

In the window mean-payoff objective, given an infinite path, instead of considering a long run average, we consider the minimum payoff that can be ensured at every position of the path over a finite window that slides over the entire path. Chatterjee et al. studied the problem to decide if in a two-player game, Player 1 has a strategy to ensure a window mean-payoff of at least 0. In this work, we consider a function that given a path returns the supremum value of the window mean-payoff that can be ensured over the path and we show how to compute its expected value in Markov chains and Markov decision processes. We consider two variants of the function: Fixed window mean-payoff in which a fixed window length $l_{max}$ is provided; and Bounded window mean-payoff in which we compute the maximum possible value of the window mean-payoff over all possible window lengths. Further, for both variants, we consider (i) a direct version of the problem where for each path, the payoff that can be ensured from its very beginning and (ii) a non-direct version that is the prefix independent counterpart of the direct version of the problem.Comment: Replaced PP-hardness of direct fixed window objective with PSPACE-hardness, added alternative definition of window mean-payof