The present paper shows how to incorporate the correlation between successive nuclear jumps, which are well known to play a role in defect mechanisms of self-diffusion in crystals, into the calculation of the correlation functions describing the temperature dependence of the nuclear spin relaxation in solids. The treatment is based on an extension of Torrey’s theory of nuclear spin relaxation by randomly migrating nuclei. Explicit and detailed results are given for the monovacancy mechanisms of self-diffusion in an isotopically pure face-centred cubic crystal.
Within road-cycling, the optimization of performance using mathematical models has primarily been performed in the individual time trial. Nevertheless, most races are 'mass-start' events in which many riders compete at the same time. In some special situations, e.g. breakaways from the peloton, the riders are forced to team up. To simulate those cooperative rides of two athletes, an extension of models and optimization approaches for individual time trials is presented. A slipstream model based on experimental data is provided to simulate the physical interaction between the two riders. In order to simulate real world behavior, a penalty for the difference in the exertion levels of the two riders is introduced. This means, that even though both riders aim to be as fast as possible as a group, neither of them should have an advantage over the other because of significantly different levels of fatigue during the ride. In our simulations, the advantage of cooperation of two equally trained athletes adds up to a time gain of about 10% compared to an individual ride.