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This chapter provides a brief review of recent developments on two nonlocal operators: fractional Laplacian and fractional time derivative. We start by accounting for several applications of these operators in imaging science, geophysics, harmonic maps, and deep (machine) learning. Various notions of solutions to linear fractional elliptic equations are provided and numerical schemes for fractional Laplacian and fractional time derivative are discussed. Special emphasis is given to exterior optimal control problems with a linear elliptic equation as constraints. In addition, optimal control problems with interior control and state constraints are considered. We also provide a discussion on fractional deep neural networks, which is shown to be a minimization problem with fractional in time ordinary differential equation as constraint. The paper concludes with a discussion on several open problems. 

This chapter aims to look at the properties of large helium nanodroplets from two different perspectives: a.) helium droplets as hosts for assembling and studying clusters at low temperatures; and b.) helium droplets as systems to be studied on their own. First, the thermodynamics and excitations in large droplets are presented, followed by a primer on the droplets’ rate of cooling in vacuum. The chapter then proceeds with the description on producing and characterising the droplets. This subject is followed by a discussion on the kinetics for different regimes of cluster aggregation, such as that for single- and multiple-centre aggregation. Then, experiments involving the spectroscopy of the foreign particles and the deposition of metallic clusters for electron microscopy studies are described. Finally, results from recent x-ray coherent diffractive imaging experiments with pure and doped helium nanodroplets are summarised.