Abstract In part I of this work, we established an analytic framework for solving Maxwell's equations in spherical geometry with continuous angular indices (ℓ,m)∈R2, proving finite-energy conditions for singular solutions and formulating continuous spectral integrals in weighted Sobolev spaces. In this companion article (part II), we develop and validate numerical methods that operationalize this framework. We introduce a Galerkin discretization of the coupled angular operator on divergence-free continuous-index vector bases. To reconstruct physical fields, we complement this with an Eisenstein-type spectral integral, evaluated by mapped Gauss–Legendre quadrature. These methods explicitly account for non-orthogonality of the basis and the singular behaviour near the origin. The block formulation acts on the full matrix-valued angular Maxwell operator and is not restricted to the wedge and cone geometries used below as analytically tractable benchmarks. We analyse and numerically demonstrate convergence. The hybrid pipeline is validated by reproducing eigenmode frequencies of spherical cavities with wedge and cone modifications to within 1−2% of full-wave high-frequency structure simulator (HFSS) simulations. Together with part I, this paper completes a unified analytic–computational framework for singular electromagnetic fields with continuous angular indices.
Abstract We consider a generic two-level quantum system with control parameters as the coupling between the two states, taking values in a bounded positive interval, and the detuning, restricted in a symmetric interval around zero. Using Stokes theorem, we prove that the optimal path for fast quasi-adiabatic dynamics lies mainly on the border of parameter space, as it was numerically found in a recent study regarding coupled elastic waveguides, modelled as a two-level system. We calculate analytically the adiabaticity parameter as a function of the protocol duration and compare with the finite-time Landau–Zener protocol, where the coupling is kept constant while the detuning varies linearly with time. The optimized-path protocol shows superior performance even for shorter durations, where the Landau–Zener protocol fails. The present study may be useful in different fields with systems modelled as the investigated generic two-level quantum system.
Abstract This article presents a systematic approach for deriving the necessary and sufficient conditions for diffusion-driven instability for a three-component reaction–diffusion system (RDS) with linear cross-diffusion and activator-bi-inhibitor reaction kinetics. First, by considering the system in the absence of spatial variations, we derive the necessary conditions for the uniform steady state to be linearly asymptotically stable, based on the Routh–Hurwitz criterion. Second, by introducing spatial variations, we derive 17 different sets of necessary conditions under which the uniform steady state becomes linearly unstable. These conditions emerge from the violation of at least one of the Routh–Hurwitz conditions for stability. The derivation of these conditions reduces to studying one quadratic and two cubic polynomials as functions of the wave numbers. This is in contrast to the case of a two-component RDS, where Turing instability is driven by a single quadratic polynomial. To verify the validity of these conditions, we systematically derive diffusion-driven instability parameter spaces by picking cross-diffusion tensor matrices from real physical systems. Sufficient conditions are established by determining parameter values from the generated parameter spaces such that specific modes are isolated for pattern formation. Finite element simulations in 1–space dimension confirm theoretical predictions of linear stability analysis.
Abstract Most birds can transition from aerial to terrestrial locomotion in complex environments. However, to touchdown at safe speeds is challenging as low speeds result in poor aerodynamic control authority, especially in unpowered flight, and how birds manage this is not well understood. Through exploration of the dynamics, we provide an analytical optimal control strategy that minimizes touchdown speed and show that zero touchdown speed is physically possible in unpowered flight. However, this comes at the cost of large approach velocities and displacements, which are not commonly observed in birds, indicating that other factors probably influence their flight behaviour. When flights are constrained to land on the ground, unpowered landings can no longer touchdown with zero speed, thus requiring increases in forelimb or hindlimb skeletomuscular mass to dissipate the remaining kinetic energy. This requirement is a physical constraint that may have shaped avian evolution. Comparing our model with experimental data on perching birds provides new evidence to support a proposal that birds minimize touchdown speed and time spent post-stall. Overall, our results provide a mechanical basis for understanding landing behaviour and highlight important physical constraints that may have affected avian evolution.
Abstract Shear bands in inelastic solids are often treated as stationary failure zones whose orientation is fixed at bifurcation, a view reinforced by classical Cauchy continua with softening and no internal length scale, where loss of ellipticity drives deformation into a single planar band. Experiments in granular media, however, increasingly show curved, branching and migrating shear zones. Here, we study finite-strain localization in a geometrically nonlinear micropolar (Cosserat) continuum implemented in an implicit material point method, using a non-associative Drucker–Prager model with a unified yield surface based on a generalized measure of stresses and couple stresses to introduce an intrinsic length scale. Two-dimensional boundary-value problems show that localization need not remain rectilinear or stationary: the active zone can curve, reorient and migrate as the stress state evolves, without ad hoc parameter tuning. Viewed through the microrotation field, the evolving centreline is consistently traced by a near-zero ridge of the band-normal microrotation gradient, which we propose as a natural band-selection rule in the nonlinear micropolar continuum.
