Scalar relative invariants play an important role in the theory of group actions on a manifold as their zero sets are invariant hypersurfaces. Relative invariants are central in many applications, where they often are treated locally since an invariant hypersurface may not be a locus of a single function. Our aim is to establish a global theory of relative invariants.

For a Lie algebra {\mathfrak g}${\mathfrak g}$ of holomorphic vector fields on a complex manifold M, any holomorphic {\mathfrak g}${\mathfrak g}$-invariant hypersurface is given in terms of a {\mathfrak g}${\mathfrak g}$-invariant divisor. This generalizes the classical notion of scalar relative {\mathfrak g}${\mathfrak g}$-invariant. Any {\mathfrak g}${\mathfrak g}$-invariant divisor gives rise to a {\mathfrak g}${\mathfrak g}$-equivariant line bundle, and a large part of this paper is therefore devoted to the investigation of the group \mathrm {Pic}_{\mathfrak g}(M)$\mathrm {Pic}_{\mathfrak g}(M)$ of {\mathfrak g}${\mathfrak g}$-equivariant line bundles. We give a cohomological description of \mathrm {Pic}_{\mathfrak g}(M)$\mathrm {Pic}_{\mathfrak g}(M)$ in terms of a double complex interpolating the Chevalley-Eilenberg complex for {\mathfrak g}${\mathfrak g}$ with the Čech complex of the sheaf of holomorphic functions on M.

We also obtain results about polynomial divisors on affine bundles and jet bundles. This has applications to the theory of differential invariants. Those were actively studied in relation to invariant differential equations, but the description of multipliers (or weights) of relative differential invariants was an open problem. We derive a characterization of them with our general theory. Examples, including projective geometry of curves and second-order ODEs, not only illustrate the developed machinery but also give another approach and rigorously justify some classical computations. At the end, we briefly discuss generalizations of this theory.

Wave impact on solid structures is a well-studied phenomenon, but almost exclusively for the case that the impacting liquid (e.g. water) is surrounded by a non-condensable gas (such as air). In this study we turn to wave impact in a boiling liquid, a liquid that is in thermal equilibrium with its own vapour, which is of key relevance to the transport of cryogenic liquids, such as liquified natural gas and liquid hydrogen in the near future. More specifically, we use the Atmosphere facility at MARIN, NL, to prepare water/water vapour systems at different temperatures along the vapour curve. Here, we perform wave impact experiments by generating a soliton in a flume contained within the autoclave of the facility. A bathymetry profile interacts with the soliton, leading to a breaking wave that impacts onto a vertical wall, where we measure the pressures occurring during impact by means of 100$100$ embedded pressure sensors. In boiling liquids, we report wave impact pressures that are up to two orders of magnitude larger than those measured in comparable water–air experiments. We trace these pressures back to the collapse of the entrapped vapour pocket, which we semi-quantitatively describe using a simplified hemicylindrical vapour bubble model, which is in good agreement with the experimental findings. Finally, this allows us to predict the relevance of our findings for the transport of cryogenic liquids in huge overseas carriers where wave impact due to sloshing is the dominant cause of hydrodynamic load of containment systems in cargo tanks.

In a quiescent medium, chemically active particles propel themselves by emitting or absorbing solutes, creating concentration gradients that induce a slip at the particle surface. This self-propulsion occurs when solute advection overcomes diffusion. However, an imposed flow field can alter these dynamics. This study explores the propulsion characteristics and the related rheological consequences of chemically active particles in an imposed uniaxial extensional flow analytically and numerically. An asymptotic solution is obtained for weak imposed flow relative to self-induced diffusiophoretic slip. Meanwhile, finite element simulations are carried out over a wide range of imposed flow strength and Péclet number. The results reveal that the interplay between solute advection, imposed flow and diffusiophoretic slip significantly affects particle propulsion and suspension rheology. While solute advection and diffusiophoretic slip tend to create asymmetric solute distributions, promoting self-propulsion, imposed extensional flow promotes symmetric distributions, hindering self-propulsion. This not only delays the start of self-propulsion but also results in an early transition from a propulsion state to a stationary state characterised by an abrupt halt at relatively lower Péclet number compared to a quiescent medium. Post the abrupt halt, a stirring effect induced by particle activity and imposed extensional flow results in an increased magnitude of stresslet, thus a sudden change in the effective viscosity of the active suspension. The effect of imposed extensional flow on active particle dynamics and suspension rheology can be described succinctly by categorising the overall dynamics into three separate regimes, determined by the Péclet number and the intensity of the extensional flow.

The acoustofluidic method holds great promise for manipulating micro-organisms. When exposed to the steady vortex structures of acoustic streaming flow, these micro-organisms exhibit intriguing dynamic behaviours, such as hydrodynamic trapping and aggregation. To uncover the mechanisms behind these behaviours, we investigate the swimming dynamics of both passive and active particles within a two-dimensional acoustic streaming flow. By employing a theoretically calculated streaming flow field, we demonstrate the existence of stable bounded orbits for particles. Additionally, we introduce rotational diffusion and examine the distribution of particles under varying flow strengths. Our findings reveal that active particles can laterally migrate across streamlines and become trapped in stable bounded orbits closer to the vortex centre, whereas passive particles are confined to movement along the streamlines. We emphasise the influence of the flow field on the distribution and trapping of active particles, identifying a flow configuration that maximises their aggregation. These insights contribute to the manipulation of microswimmers and the development of innovative biological microfluidic chips.

The conditional expectation m_{X}(s)=\mathrm{E}[X|S=s]$m_{X}(s)=\mathrm{E}[X|S=s]$, where X and Y are two independent random variables with S=X+Y$S=X+Y$, plays a key role in various actuarial applications. For instance, considering the conditional mean risk-sharing rule, m_X(s)$m_X(s)$ determines the contribution of the agent holding the risk X to a risk-sharing pool. It is also a relevant function in the context of risk management, for example, when considering natural capital allocation principles. The monotonicity of m_X(\!\cdot\!)$m_X(\!\cdot\!)$ is particularly significant under these frameworks, and it has been linked to log-concave densities since Efron (1965). However, the log-concavity assumption may not be realistic in some applications because it excludes heavy-tailed distributions. We consider random variables with regularly varying densities to illustrate how heavy tails can lead to a nonmonotonic behavior for m_X(\!\cdot\!)$m_X(\!\cdot\!)$. This paper first aims to identify situations where m_X(\!\cdot\!)$m_X(\!\cdot\!)$ could fail to be increasing according to the tail heaviness of X and Y. Second, the paper aims to study the asymptotic behavior of m_X(s)$m_X(s)$ as the value s of the sum gets large. The analysis is then extended to zero-augmented probability distributions, commonly encountered in applications to insurance, and to sums of more than two random variables and to two random variables with a Farlie–Gumbel–Morgenstern copula. Consequences for risk sharing and capital allocation are discussed. Many numerical examples illustrate the results.