The traditional ground collision avoidance system (GCAS) makes avoidance decisions based on predicted collision time, without considering the impact of terrain environment and dynamic changes in load factor on avoidance decisions. This increases the risk of ground collisions for the aircraft. To solve the problem, a GCAS with multi-trajectory risk assessment and decision function is proposed. Firstly, a variety of predicted flight avoidance trajectories are established within the final manoeuvering capability of the aircraft. Secondly, for each predicted trajectory, the uncertain length between adjacent prediction points is used to construct a rectangular distance bin, and the terrain data below the avoided trajectory is extracted. Finally, the regret theory is used to establish a multi-attribute avoidance decision model to evaluate and prioritise the risk of collision avoidance trajectories, to provide effective collision avoidance decision for pilots. The algorithm is tested and verified with real digital elevation model and simulated flight data, and compared with traditional GCAS. Simulation results show that the proposed algorithm can comprehensively consider manoeuvering performance and threatening terrain, and provide safe and effective avoidance decisions for pilots.

We discuss a continuous-time Markov branching model in which each individual can trigger an alarm according to a Poisson process. The model is stopped when a given number of alarms is triggered or when there are no more individuals present. Our goal is to determine the distribution of the state of the population at this stopping time. In addition, the state distribution at any fixed time is also obtained. The model is then modified to take into account the possible influence of death cases. All distributions are derived using probability-generating functions, and the approach followed is based on the construction of families of martingales.

The importance of long-distance alignments in Roman surveying is increasingly being recognised. It has now been discovered that they were used in setting out the central sectors of the Antonine Wall, but — in contrast to Hadrian's Wall — it appears that they were employed to determine the locations of the military installations along the Wall rather than the line of its rampart and ditch. It also appears that the enigmatic enclosures and expansions which are attached to the rear of the rampart of the Wall only seem to occur in connection with these alignments. A careful analysis of possible explanations indicates that the Romans may initially have sought to set up a two-level alarm system across the central sectors of the Antonine Wall, the possible impacts of which upon the planning and design of the Wall are examined. The Supplementary Material available online (https://doi.org/10.1017/S0068113X18000284) contains a table of inter-visabilities between known military installations, and OS grid references for the installations.