We determine all CR maps from the sphere in C3 into the tube over the future light cone in C4. This result leads to a complete characterization of proper holomorphic maps from the three-dimensional unit ball into the classical domain of type IV of dimension four, confirms a conjecture of Reiter–Son from 2022, and settles the case left open in Xiao–Yuan [36] and Reiter–Son [29]. Additionally, we prove a boundary characterization of isometric holomorphic embeddings from a ball into a classical domain of type IV in arbitrary dimensions that is similar to the main result in Huang–Lu–Tang–Xiao [18]. The result is then used to treat a special case in the general characterization.
In 1982, Schlickewei and Van der Poorten claimed that any multi-recurrence sequence has, essentially, maximal possible growth rate. Fourty years later, Fuchs and Heintze provided a non-effective proof of this statement. In this paper, we prove a quantitative version of that result by giving an explicit upper bound for the maximal possible growth rate of a multi-recurrence. Moreover, we also give a function field analogue of the result, answering a question posed by Fuchs and Heintze when proving a bound on the growth of multi-recurrences in number fields.
We address the numerical solution of second-order Mean Field Game problems through Newton iterations in infinite dimensions, introduced in [1], where quadratic convergence of the method was rigorously established. Building upon this theoretical framework, we develop new numerical discretization techniques, including both a finite difference and a semi-Lagrangian scheme, that enable an effective computational implementation of the infinite-dimensional iterations. The proposed methods are tested on several benchmark problems, and the resulting numerical experiments demonstrate their robustness, accuracy, and efficiency. A comparative analysis between the two schemes and existing approaches from the literature is also presented, highlighting the potential of Newton-based solvers for MFG systems.
We study a diffuse-interface model for thermally driven phase separation in viscous incompressible mixtures. The system couples a convective Cahn-Hilliard equation for the order parameter with a Stokes subsystem for the velocity-pressure field and a heat equation for the temperature. Temperature enters the bulk free energy through a Landau-type coefficient, while the phase field affects the flow through concentration-dependent density and viscosity. The model serves as a proxy for temperature-triggered condensation-like phase separation; humidity, latent heat, vapor pressure, and capillary forcing are absorbed into the choice of the threshold temperature Θ_S. We motivate the chemical potential through a temperature-dependent Landau free energy and use a regularized auxiliary formulation to prove local-in-time existence of weak solutions. For the numerical analysis, we employ a first-order sequential finite-element discretization of a simplified quasi-static formulation. The heat equation is advanced by implicit diffusion, the variable-coefficient Stokes problem is treated by a Taylor-Hood discretization, and the Cahn-Hilliard bulk derivative is evaluated at the previous time level, so each algebraic subproblem is linear. An isothermal diffusive test confirms mass conservation to roundoff and exhibits monotone discrete-energy decay for the tested parameters. Time-step and mesh-refinement studies show first-order temporal and approximately second-order spatial behavior. The remaining computations provide qualitative, parameter-specific illustrations; no global discrete energy law is claimed for the non-isothermal sequential scheme.
Urbanisation is a globally increasing phenomenon with diverse impacts on biodiversity. Interest in how urbanisation affects raptors is growing because, as top predators, they can be used as bioindicators for ecosystem functioning and sentinels for environmental change. However, a comprehensive synthesis detailing the global-scale impact of urban-related sensory and risk factors on nocturnal raptors (i.e. owls) is lacking. In this review, we examined the literature to identify such factors and to outline their association with behavioural and ecological traits of owls living in urban environments. Overall, we show that several urban-related sensory and risk factors affect owls, with vehicle collisions on roads being the most widely documented across species. Conversely, sensory pollution remains poorly investigated, which is surprising given that nocturnal and acoustic hunters, such as owls, might be severely impacted by artificial light at night (ALAN) and anthropogenic noise. We also highlight a research gap on this topic from the global south, where urbanisation is rapidly increasing. Importantly, we show that roads and sensory pollutants are associated in contrasting ways with many owl behavioural and ecological traits, such as hunting and habitat use. We argue that the interplay among roads, noise and ALAN influences how owls’ prey species use roads, which may turn areas near roads into ecological traps. The severity of their impacts may depend on the intensity and type of anthropogenic noise and artificial lights along roads. Further research in this direction will have important implications for the conservation of owls in urban environments.
