No abstract

We characterize the set of extreme points of monotonic functions that are either majorized by a given function f or themselves majorize f and show that these extreme points play a crucial role in many economic design problems. Our main results show that each extreme point is uniquely characterized by a countable collection of intervals. Outside these intervals the extreme point equals the original function f and inside the function is constant. Further consistency conditions need to be satisfied pinning down the value of an extreme point in each interval where it is constant. We apply these insights to a varied set of economic problems: equivalence and optimality of mechanisms for auctions and (matching) contests, Bayesian persuasion, optimal delegation, and decision making under uncertainty.

Non-axisymmetric free-boundary equilibrium computations are shown to represent nonlinearly saturated external kink modes and external kink-like sidebands coupled to pressure-driven infernal modes. In this study of edge harmonic oscillations associated with QH-mode plasmas, two different driving mechanisms for external kink type-modes are identified. It is found that standard current-driven external kinks are linearly unstable, and nonlinearly stable in a wide parameter range, especially where q edge  m/n. But, where standard current-driven kinks are linearly stable coupling of pressure-driven infernal modes can cause instability, and their upper sideband drives edge corrugations that appear to have external kink features. Both types of modes are identified with the VMEC equilibrium code, and the spectra are compared favourably with those of linear numerical approaches and analytic methods. Pressure-driven external infernal modes are shown to robustly occur in sophisticated modelling where the separatrix effect on the q profile is accounted for.

It is shown that free boundary 3D equilibrium calculations in tokamak geometry are capable of capturing the physics of non-linearly saturated external kink modes for monotonic current and q profiles typical of standard (baseline) plasma scenarios. The VMEC ideal MHD equilibrium model exhibits strong flux surface corrugations of the plasma vacuum boundary, driven by the core current profile. A method is presented which conveniently extracts the amplitude of the corrugation in terms of Fourier components in straight field line coordinates. The Fourier spectrum, and condition for non-linear corrugation agrees well with linear simulations, and the saturated amplitude agrees well with non-linear analytic calculations.

We explore the impact of realistic plasma resistivity on the linear stability of peeling-ballooning (PB) modes in tokamak discharges with low-aspect ratio. For this study we consider discharges that are subject to edge-localized modes (ELMs) in the National Spherical Torus Experiment (NSTX). Employing the state of the art extended-magnetohydrodynamic (MHD) code M3D-C1 it is demonstrated that non-ideal effects can significantly affect PB stability thresholds in NSTX discharges. In particular, robust resistive PB modes are found to exist well before the ideal PB stability threshold is met. These novel results can explain why ideal-MHD theory often does not accurately describe ELM onset in spherical torus configurations, and also present a valuable basis for the development of a predictive model for ELMs in low-aspect ratio tokamaks.