In this paper, we attempt to define and understand the orbits of the Koch snowflake fractal billiard KS. This is a priori a very difficult problem because ∂(KS), the snowflake curve boundary of KS, is nowhere differentiable, making it impossible to apply the usual law of reflection at any point of the boundary of the billiard table. Consequently, we view the prefractal billiards KSn (naturally approximating KS from the inside) as rational polygonal billiards and examine the corresponding flat surfaces of KSn, denoted by S KSn . In order to develop a clearer picture of what may possibly be happening on the billiard KS, we simulate billiard trajectories on KSn (at first, for a fixed n ≥ 0). Such computer experiments provide us with a wealth of questions and lead us to formulate conjectures about the existence and the geometric properties of periodic orbits of KS and detail a possible plan on how to prove such conjectures.

Abstract. If D is a rational polygon, then the associated rational billiard table is given by Ω(D). Such a billiard table is well understood. If F is a closed fractal curve approximated by a sequence of rational polygons, then the corresponding fractal billiard table is denoted by Ω(F ). In this paper, we survey many of the results from [LapNie1-3] for the Koch snowflake fractal billiard Ω(KS) and announce new results on two other fractal billiard tables, namely, the T -fractal billiard table Ω(T ) (see [LapNie6]) and a self-similar Sierpinski carpet billiard table Ω(Sa) (see [CheNie]).We build a general framework within which to analyze what we call a sequence of compatible orbits. Properties of particular sequences of compatible orbits are discussed for each prefractal billiard Ω(KSn), Ω(Tn) and Ω(Sa,n), for n = 0, 1, 2 · · · . In each case, we are able to determine a particular limiting behavior for an appropriately formulated sequence of compatible orbits. Such a limit either constitutes what we call a nontrivial path of a fractal billiard table Ω(F ) or else a periodic orbit of Ω(F ) with finite period. In our examples, F will be either KS, T or Sa. Several of the results and examples discussed in this paper are presented for the first time.We then close with a brief discussion of open problems and directions for further research in the emerging field of fractal billiards.

We identify a collection of periodic billiard orbits in a self-similar Sierpinski carpet billiard table Ω(Sa). Based on a refinement of the result of Durand-Cartagena and Tyson regarding nontrivial line segments in Sa, we construct what is called an eventually constant sequence of compatible periodic orbits of prefractal Sierpinski carpet billiard tables Ω(Sa,n). The trivial limit of this sequence then constitutes a periodic orbit of Ω(Sa). We also determine the corresponding translation surface S(Sa,n) for each prefractal table Ω(Sa,n), and show that the genera {gn} ∞ n=0 of a sequence of translation surfaces {S(Sa,n)} ∞ n=0 increase without bound. Various open questions and possible directions for future research are offered.

Abstract. Twenty years ago a theoretical analysis showed that electron scattering by high-Z one-electron atoms might lead to interference of exchange and spin-orbit interactions at low electron energies, observable as a crosssection asymmetry if unpolarized electrons are scattered by polarized cesium atoms. By using a highly polarized cesium atomic beam, we studied exchange and spin-orbit effects at different electron energies, starting at 20 eV and going down. We observed the first distinctly non-zero interference asymmetry at 7 eV: Over the angular range of 35 to 145 ° , it varies between +0.02 and -0.02 and goes through zero near 110 ° being negative at larger angles.

This study uses Bayesian simulations to estimate the probability that published criminological research findings are wrong. Toward this end, we employ two equations originally popularized in John P.A. Ioannidis’ (in)famous article, “Why Most Published Research Findings are False.” Values for relevant parameters were determined using recent estimates for the field’s average level of statistical power, level of research bias, level of factionalization, and quality of theory. According to our simulations, there is a very high probability that most published criminological research findings are false-positives, and therefore wrong. Further, we demonstrate that the primary factor contributing to this problem is the poor quality of theory. Stated differently, even when the overall level of research bias is extremely low and overall statistical power is extremely high, we find that poor theory still results in a high rate of false positives. We conclude with suggestions for improving the validity of criminological research claims.