On the Existence of Minimal Hypersurfaces with Arbitrarily Large Area and Morse Index

We show that a bumpy closed Riemannian manifold $(M^{n+1}, g)$ $(3 \leq n+1 \leq 7)$ admits a sequence of connected closed embedded two-sided minimal hypersurfaces whose areas and Morse indices both tend to infinity. This improves a previous result by O. Chodosh and C. Mantoulidis on connected minimal hypersurfaces with arbitrarily large area.Comment: Final version, to appear in Geom. Topo

Integer colorings with forbidden rainbow sums

For a set of positive integers $A \subseteq [n]$, an $r$-coloring of $A$ is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erd\H{o}s-Rothchild problem in the context of sum-free sets, which asks for the subsets of $[n]$ with the maximum number of rainbow sum-free $r$-colorings. We show that for $r=3$, the interval $[n]$ is optimal, while for $r\geq8$, the set $[\lfloor n/2 \rfloor, n]$ is optimal. We also prove a stability theorem for $r\geq4$. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.Comment: 20 page

LOTUS: A (non-)LTE Optimization Tool for Uniform derivation of Stellar atmospheric parameters

Precise fundamental atmospheric stellar parameters and abundance determination of individual elements in stars are important for all stellar population studies. Non-Local Thermodynamic Equilibrium (Non-LTE; hereafter NLTE) models are often important for such high precision, however, can be computationally complex and expensive, which renders the models less utilized in spectroscopic analyses. To alleviate the computational burden of such models, we developed a robust 1D, LTE and NLTE fundamental atmospheric stellar parameter derivation tool, $\texttt{LOTUS}$, to determine the effective temperature $T_{\mathrm{eff}}$, surface gravity $\log g$, metallicity \mbox{[Fe/H]} and microturbulent velocity $v_{\mathrm{mic}}$ for FGK type stars, from equivalent width (EW) measurements of Fe I and Fe II lines. We utilize a generalized curve of growth method to take into account the EW dependencies of each Fe I and Fe II line on the corresponding atmospheric stellar parameters. A global differential evolution optimization algorithm is then used to derive the optimized fundamental parameters. Additionally, $\texttt{LOTUS}$ can determine precise uncertainties for each stellar parameter using a Markov Chain Monte Carlo (MCMC) algorithm. We test and apply $\texttt{LOTUS}$ on a sample of benchmark stars, as well as stars with available asteroseismic surface gravities from the K2 survey, and metal-poor stars from $R$-process Alliance (RPA) survey. We find very good agreement between our NLTE-derived parameters in $\texttt{LOTUS}$ to non-spectroscopic values within $T_{\mathrm{eff}}=\pm 30$ K and $\log g=\pm 0.10$ dex for benchmark stars. We provide open access of our code, as well as of the interpolated pre-computed NLTE EW grids available on Github, and documentation with working examples on Readthedocs.Comment: 42 pages, 12 figures, 3 tables, accepted for publication in AJ. For Code on Github, see: https://github.com/Li-Yangyang/LOTUS. For Code Documentation, see: https://lotus-nlte.readthedocs.io/en/latest