In Search of a Fair Bet in the Lottery

Although state-operated lotto games have the worst average expected payoffs among common games of chance, because the jackpot can accumulate, the maximum expected payoff is potentially unlimited. It is possible, therefore, that lotto can exhibit a positive expected return. This paper examines 18,000 drawings in 34 American lotteries and finds approximately 1 percent of these drawings provided players with a fair bet. If it were possible for a bettor to purchase every possible combination, however, most lotteries commonly experience circumstances where such a purchase would provide a positive return with 11 percent of the drawings providing a fair bet to the player.

Amplified Sensitivity of Nitrogen-Vacancy Spins in Nanodiamonds using All-Optical Charge Readout

Nanodiamonds containing nitrogen-vacancy (NV) centers offer a versatile platform for sensing applications spanning from nanomagnetism to in-vivo monitoring of cellular processes. In many cases, however, weak optical signals and poor contrast demand long acquisition times that prevent the measurement of environmental dynamics. Here, we demonstrate the ability to perform fast, high-contrast optical measurements of charge distributions in ensembles of NV centers in nanodiamonds and use the technique to improve the spin readout signal-to-noise ratio through spin-to-charge conversion. A study of 38 nanodiamonds, each hosting 10-15 NV centers with an average diameter of 40 nm, uncovers complex, multiple-timescale dynamics due to radiative and non-radiative ionization and recombination processes. Nonetheless, the nanodiamonds universally exhibit charge-dependent photoluminescence contrasts and the potential for enhanced spin readout using spin-to-charge conversion. We use the technique to speed up a $T_1$ relaxometry measurement by a factor of five.Comment: 13 pages, 14 figure

Modeling laser wakefield accelerators in a Lorentz boosted frame

Modeling of laser-plasma wakefield accelerators in an optimal frame of reference \cite{VayPRL07} is shown to produce orders of magnitude speed-up of calculations from first principles. Obtaining these speedups requires mitigation of a high-frequency instability that otherwise limits effectiveness in addition to solutions for handling data input and output in a relativistically boosted frame of reference. The observed high-frequency instability is mitigated using methods including an electromagnetic solver with tunable coefficients, its extension to accomodate Perfectly Matched Layers and Friedman's damping algorithms, as well as an efficient large bandwidth digital filter. It is shown that choosing the frame of the wake as the frame of reference allows for higher levels of filtering and damping than is possible in other frames for the same accuracy. Detailed testing also revealed serendipitously the existence of a singular time step at which the instability level is minimized, independently of numerical dispersion, thus indicating that the observed instability may not be due primarily to Numerical Cerenkov as has been conjectured. The techniques developed for Cerenkov mitigation prove nonetheless to be very efficient at controlling the instability. Using these techniques, agreement at the percentage level is demonstrated between simulations using different frames of reference, with speedups reaching two orders of magnitude for a 0.1 GeV class stages. The method then allows direct and efficient full-scale modeling of deeply depleted laser-plasma stages of 10 GeV-1 TeV for the first time, verifying the scaling of plasma accelerators to very high energies. Over 4, 5 and 6 orders of magnitude speedup is achieved for the modeling of 10 GeV, 100 GeV and 1 TeV class stages, respectively