In this paper, the third in the series, we continue our study of combinatorics in chaotic Newtonian dynamics. We study the chaotic four-body problem in Newtonian gravity assuming finite-sized particles, and we focus on interactions that produce direct collisions between any two stars. Our long-term goal is to construct an equation that gives the probability of a given collision event occurring over the course of the interaction, as a function of the total encounter energy and angular momentum as well as the numbers and properties of the particles. In previous papers, we varied the number of interacting particles and the distribution of particle radii, for all equal mass particles. Here, we focus on the effects of different combinations of particle masses. We develop an analytic formalism for calculating the time-scales for different collision scenarios to occur. Our analytic time-scales reproduce the simulated time-scales when gravitational focusing is included. We present a method for calculating the relative rates for different types of collisions to occur, assuming two different limits for the particle orbits; radial and tangential. These limits yield relative collision probabilities that bracket the probabilities we obtain directly from numerical scattering experiments, and are designed to reveal important information about the (time-averaged) trajectories of the particles as a function of the interaction parameters. Finally, we present a Collision Rate Diagram (CRD), which directly compares the predictions of our analytic rates to the simulations and quantifies the quality of the agreement. The CRD will facilitate refining our analytic collision rates in future work, as we expand in to the remaining parameter space.
World record setting has long attracted public interest and scientific investigation. Extremal records summarize the limits of the space explored by a process, and the historical progression of a record sheds light on the underlying dynamics of the process. Existing analyses of prediction, statistical properties, and ultimate limits of record progressions have focused on particular domains. However, a broad perspective on how record progressions vary across different spheres of activity needs further development. Here we employ cross-cutting metrics to compare records across a variety of domains, including sports, games, biological evolution, and technological development. We find that these domains exhibit characteristic statistical signatures in terms of rates of improvement, "burstiness" of record-breaking time series, and the acceleration of the record breaking process. Specifically, sports and games exhibit the slowest rate of improvement and a wide range of rates of "burstiness." Technology improves at a much faster rate and, unlike other domains, tends to show acceleration in records. Many biological and technological processes are characterized by constant rates of improvement, showing less burstiness than sports and games. It is important to understand how these statistical properties of record progression emerge from the underlying dynamics. Towards this end, we conduct a detailed analysis of a particular record-setting event: elite marathon running. In this domain, we find that studying record-setting data alone can obscure many of the structural properties of the underlying process. The marathon study also illustrates how some of the standard statistical assumptions underlying record progression models may be inappropriate or commonly violated in real-world datasets.
NORMA is a design for a normal-conducting race track fixed-field alternating-gradient accelerator (FFAG) for protons from 50 to 350 MeV. In this article we show the development from an idealised lattice to a design implemented with field maps from rigorous two-dimensional (2D) and three-dimensional (3D) FEM magnet modelling. We show that whilst the fields from a 2D model may reproduce the idealised field to a close approximation, adjustments must be made to the lattice to account for differences brought about by the 3D model and fringe fields and full 3D models. Implementing these lattice corrections we recover the required properties of small tune shift with energy and a sufficiently-large dynamic aperture. The main result is an iterative design method to produce the first realistic design for a proton therapy accelerator that can rapidly deliver protons for both treatment and for imaging at up to 350 MeV. The first iteration is performed explicitly and described in detail in the text.
Orbital period in a ring accelerator and time of flight in a linear accelerator depend on the amplitude of betatron oscillations. The variation is negligible in ordinary particle accelerators with relatively small beam emittance. In an accelerator for large emittance beams like muons and unstable nuclei, however, this effect cannot be ignored. We measured orbital period in a linear non-scaling fixed field alternating gradient (FFAG) accelerator, which is a candidate for muon acceleration, and compared with the theoretical prediction. The good agreement between them gives important ground for the design of particle accelerators for a new generation of particle and nuclear physics experiments.
Delay-coordinate reconstruction is a proven modeling strategy for building effective forecasts of nonlinear time series. The first step in this process is the estimation of good values for two parameters, the time delay and the embedding dimension. Many heuristics and strategies have been proposed in the literature for estimating these values. Few, if any, of these methods were developed with forecasting in mind, however, and their results are not optimal for that purpose. Even so, these heuristics---intended for other applications---are routinely used when building delay coordinate reconstruction-based forecast models. In this paper, we propose a new strategy for choosing optimal parameter values for forecast methods that are based on delay-coordinate reconstructions. The basic calculation involves maximizing the shared information between each delay vector and the future state of the system. We illustrate the effectiveness of this method on several synthetic and experimental systems, showing that this metric can be calculated quickly and reliably from a relatively short time series, and that it provides a direct indication of how well a near-neighbor based forecasting method will work on a given delay reconstruction of that time series. This allows a practitioner to choose reconstruction parameters that avoid any pathologies, regardless of the underlying mechanism, and maximize the predictive information contained in the reconstruction.
Community detection in online social networks is typically based on the analysis of the explicit connections between users, such as "friends" on Facebook and "followers" on Twitter. But online users often have hundreds or even thousands of such connections, and many of these connections do not correspond to real friendships or more generally to accounts that users interact with. We claim that community detection in online social networks should be question-oriented and rely on additional information beyond the simple structure of the network. The concept of 'community' is very general, and different questions such as "whom do we interact with?" and "with whom do we share similar interests?" can lead to the discovery of different social groups. In this paper we focus on three types of communities beyond structural communities: activity-based, topic-based, and interaction-based. We analyze a Twitter dataset using three different weightings of the structural network meant to highlight these three community types, and then infer the communities associated with these weightings. We show that the communities obtained in the three weighted cases are highly different from each other, and from the communities obtained by considering only the unweighted structural network. Our results confirm that asking a precise question is an unavoidable first step in community detection in online social networks, and that different questions can lead to different insights about the network under study.
The next generation of lepton flavor violation experiments need high intensity and high quality muon beams. Production of such beams requires sending a short, high intensity proton pulse to the pion production target, capturing pions and collecting the resulting muons in the large acceptance transport system. The substantial increase of beam quality can be obtained by applying the RF phase rotation on the muon beam in the dedicated FFAG ring, which was proposed for the PRISM project.This allows to reduce the momentum spread of the beam and to purify from the unwanted components like pions or secondary protons. A PRISM Task Force is addressing the accelerator and detector issues that need to be solved in order to realize the PRISM experiment. The parameters of the required proton beam, the principles of the PRISM experiment and the baseline FFAG design are introduced. The spectrum of alternative designs for the PRISM FFAG ring are shown. Progress on ring main systems like injection and RF are presented. The current status of the study and its future directions are discussed.