Symbolic Powers of Monomial Ideals

We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal $I$ in $k[x_0, \ldots, x_n]$ we show $I^{t(m+e-1)-e+r)}$ is a subset of $M^{(t-1)(e-1)+r-1}(I^{(m)})^t$ for all positive integers $m$, $t$ and $r$, where $e$ is the big-height of $I$ and $M = (x_0, \ldots, x_n)$. This captures two conjectures ($r=1$ and $r=e$): one of Harbourne-Huneke and one of Bocci-Cooper-Harbourne. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals.Comment: 15 pages. Fixed typ

Fast forward modeling of neutral beam injection and halo formation including full Balmer-α emission prediction at W7-X

A full collisional-radiative (CR) neutral beam injection model based on Gaussian pencil (Gausscil) beams and a diffusive CR neutral halo model are presented. The halo is a neutral cloud around the neutral beam forming due to multiple charge exchange (CX) reactions. Both models do not rely on Monte-Carlo techniques and are thereby orders of magnitude faster than commonly used models. To model the neutral halo a system of coupled diffusion equations is solved numerically, enforcing mixed boundary conditions. From the equilibrium hydrogen neutral densities in the second excited energy state (n = 3), the Balmer-α emission intensity is calculated and the full spectrum is predicted, including effects as Doppler shifts and broadening due to the complex neutral beam geometry and the motional Stark effect (MSE) from the magnetic field. All forward models are implemented in the Minerva [1] Bayesian analysis framework to enable detailed multivariant inference from Balmer-α spectroscopy data. The modeled neutral beam and halo densities are successfully verified against calculations with a validated Monte-Carlo code for the W7-X beam and plasma geometry, especially proving the validity of the halo diffusion ansatz. A comparison of the predicted emission spectra with the experimental data proves the accuracy of the implemented model. All important parameters defining the neutral beams are inferred and compared to available reference values

Particle transport in reduced turbulence neutral beam heated discharges at Wendelstein 7-X

A spontaneous reduction in anomalous particle transport in the plasma core is seen experimentally in reproducible, purely neutral beam heated plasma phases at Wendelstein 7-X (W7-X). Heating and fueling the plasma exclusively with the neutral beam injection system for several seconds leads to continuously peaking plasma density profiles with strong gradients inside mid minor radius. A significant acceleration of the density peaking occurs after a certain onset time and is examined with a detailed particle transport analysis in several discharges. By invoking the particle continuity equation, the total experimental radial electron flux is deduced from the time evolution of the electron density profile and the radially resolved particle sources. Subtracting the modeled neoclassical particle flux contribution gives the anomalous particle flux. Exploiting the evolving plasma conditions, anomalous diffusion and convection coefficients are computed from the flux variation with density and density gradients. In several discharges a significant and consistent change of the anomalous transport coefficients is seen when crossing a specific normalized density gradient length

Bayesian inference of electron density and ion temperature profiles from neutral beam and halo Balmer-α emission at Wendelstein 7-X

By employing Bayesian inference techniques, the full electron density profile from the plasma core to the edge of Wendelstein 7-X (W7-X) is inferred solely from neutral hydrogen beam and halo Balmer-α (Hα) emission data. The halo is a cloud of neutrals forming in the vicinity of the injected neutral beam due to multiple charge exchange reactions. W7-X is equipped with several neutral hydrogen beam heating sources and an Hα spectroscopy system that views these sources from different angles and penetration depths in the plasma. As the beam and halo emission form complex spectra for each spatial point that are non-linearly dependent on the plasma density profile and other parameters, a complete model from the neutral beam injection and halo formation through to the spectroscopic measurements is required. The model is used here to infer electron density profiles for a range of common W7-X plasma scenarios. The inferred profiles show good agreement with profiles determined by the Thomson scattering and interferometry diagnostics across a broad range of absolute densities without any changes to the input or fitting parameters. The time evolution of the density profile in a discharge with continuous core density peaking is successfully reconstructed, demonstrating sufficient spatial resolution to infer strongly shaped profiles. Furthermore, it is shown as a proof of concept that the model is also able to infer the main ion temperature profile using the same data set