https://www.selleckchem.com/products/enpp-1-in-1.html A nematic liquid crystal confined to the surface of a sphere exhibits topological defects of total charge +2 due to the topological constraint. #link# In equilibrium, the nematic field forms four +1/2 defects, located at the corners of a regular tetrahedron inscribed within the sphere, since this minimizes the Frank elastic energy. If additionally the individual nematogens exhibit self-driven directional motion, the resulting active system creates large-scale flow that drives it out of equilibrium. In particular, the defects now follow complex dynamic trajectories which, depending on the strength of the active forcing, can be periodic (for weak forcing) or chaotic (for strong forcing). In this paper we derive an effective particle theory for this system, in which the topological defects are the degrees of freedom, whose exact equations of motion we subsequently determine. Numerical solutions of these equations confirm previously observed characteristics of their dynamics and clarify the role played by the time dependence of their global rotation. We also show that Onsager's variational principle offers an exceptionally transparent way to derive these dynamical equations, and we explain the defect mobility at the hydrodynamics level.We analyze large deviations of time-averaged quantities in stochastic processes with long-range memory, where the dynamics at time t depends itself on the value q_t of the time-averaged quantity. First consider the elephant random walk and a Gaussian variant of this model, identifying two mechanisms for unusual fluctuation behavior, which differ from the Markovian case. In particular, the memory can lead to large-deviation principles with reduced speeds and to nonanalytic rate functions. We then explain how the mechanisms operating in these two models are generic for memory-dependent dynamics and show other examples including a non-Markovian simple exclusion process.Stratification due