Separating invariants for the basic G_a-actions

We explicitly construct a finite set of separating invariants for the basic \Ga-actions. These are the finite dimensional indecomposable rational linear representations of the additive group \Ga of a field of characteristic zero, and their invariants are the kernel of the Weitzenb\"ock derivation $D_{n}=x_{0}\frac{\partial}{\partial{x_{1}}}+...+ x_{n-1}\frac{\partial}{\partial{x_{n}}}$.Comment: 10 page

The Brownian fan

We provide a mathematical study of the modified Diffusion Monte Carlo (DMC) algorithm introduced in the companion article \cite{DMC}. DMC is a simulation technique that uses branching particle systems to represent expectations associated with Feynman-Kac formulae. We provide a detailed heuristic explanation of why, in cases in which a stochastic integral appears in the Feynman-Kac formula (e.g. in rare event simulation, continuous time filtering, and other settings), the new algorithm is expected to converge in a suitable sense to a limiting process as the time interval between branching steps goes to 0. The situation studied here stands in stark contrast to the "na\"ive" generalisation of the DMC algorithm which would lead to an exponential explosion of the number of particles, thus precluding the existence of any finite limiting object. Convergence is shown rigorously in the simplest possible situation of a random walk, biased by a linear potential. The resulting limiting object, which we call the "Brownian fan", is a very natural new mathematical object of independent interest.Comment: 53 pages, 2 figures. Formerly 2nd part of arXiv:1207.286

Probing correlations of early magnetic fields using mu-distortion

The damping of a non-uniform magnetic field between the redshifts of about $10^4$ and $10^6$ injects energy into the photon-baryon plasma and causes the CMB to deviate from a perfect blackbody spectrum, producing a so-called $\mu$-distortion. We can calculate the correlation $\langle\mu T\rangle$ of this distortion with the temperature anisotropy $T$ of the CMB to search for a correlation $\langle B^2\zeta\rangle$ between the magnetic field $B$ and the curvature perturbation $\zeta$; knowing the $\langle B^2\zeta\rangle$ correlation would help us distinguish between different models of magnetogenesis. Since the perturbations which produce the $\mu$-distortion will be much smaller scale than the relevant density perturbations, the observation of this correlation is sensitive to the squeezed limit of $\langle B^2\zeta\rangle$, which is naturally parameterized by $b_{\text{NL}}$ (a parameter defined analogously to $f_{\text{NL}}$). We find that a PIXIE-like CMB experiments has a signal to noise $S/N\approx 1.0 \times b_{\text{NL}} (\tilde B_\mu/10\text{ nG})^2$, where $\tilde B_\mu$ is the magnetic field's strength on $\mu$-distortion scales normalized to today's redshift; thus, a 10 nG field would be detectable with $b_{\text{NL}}=\mathcal{O}(1)$. However, if the field is of inflationary origin, we generically expect it to be accompanied by a curvature bispectrum $\langle\zeta^3\rangle$ induced by the magnetic field. For sufficiently small magnetic fields, the signal $\langle B^2 \zeta\rangle$ will dominate, but for $\tilde B_\mu\gtrsim 1$ nG, one would have to consider the specifics of the inflationary magnetogenesis model. We also discuss the potential post-magnetogenesis sources of a $\langle B^2\zeta\rangle$ correlation and explain why there will be no contribution from the evolution of the magnetic field in response to the curvature perturbation.Comment: 23 pages, 1 figure. v2: Noted that a competing effect could potentially be smaller than originally stated. Fixed references. Matches JCAP versio

Experimental Evaluation of Book Drawing Algorithms

A $k$-page book drawing of a graph $G=(V,E)$ consists of a linear ordering of its vertices along a spine and an assignment of each edge to one of the $k$ pages, which are half-planes bounded by the spine. In a book drawing, two edges cross if and only if they are assigned to the same page and their vertices alternate along the spine. Crossing minimization in a $k$-page book drawing is NP-hard, yet book drawings have multiple applications in visualization and beyond. Therefore several heuristic book drawing algorithms exist, but there is no broader comparative study on their relative performance. In this paper, we propose a comprehensive benchmark set of challenging graph classes for book drawing algorithms and provide an extensive experimental study of the performance of existing book drawing algorithms.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Some Asymptotic Results in Discounted Repeated Games of One-Sided Incomplete Information

The paper analyzes the Nash equilibria of two-person discounted repeated games with one-sided incomplete information and known own payo®s. If the informed player is arbitrarily patient relative to the uninformed player, then the characterization for the informed player's payoffs is essentially the same as that in the undiscounted case. This implies that even small amounts of incomplete information can lead to a discontinuous change in the equilibrium payoff set. For the case of equal discount factors, however, and under an assumption that strictly individually rational payoffs exist, a result akin to the Folk Theorem holds when a complete information game is perturbed by a small amount of incomplete information.Reputation, Folk Theorem, repeated games, incomplete information.