We present photoionization models accounting for both photoelectric and resonant absorption. We demonstrate that resonance absorption lines are detectable even in moderate resolution X-ray spectra of type 1 AGN. The spectra transmitted by gas illuminated by the ionizing continua typical of flat X-ray spectrum, broad optical emission line type 1 AGN, and steep X-ray spectrum, narrow optical emis...

We consider a set X of n points in the plane, not all in a line, and the set of lines L spanned by X, where we say that a line is spanned by X if it passes through at least two elements of X. For a point P of X, we denote by t(P ) the number of lines in L which are incident to P , and call it the incident-line-number of P . We define t(X) to be MaxP∈Xt(P ), and call it the maximum incident-line...

We prove continuous symmetry breaking in three dimensions for a special class of disordered models described by the Nishimori line. The spins take values group such as $\mathbb{S}^1$, $SU(n)$ or $SO(n)$. Our proof is based on theorem about synchronization proved Abbe, Massouli\'e, Montanari, Sly and Srivastava [AMM+18]. It also relies gauge transformation acting jointly disorder spin configurat...

Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because tropical metric some nice properties such as convexity in terms of metric. One challenges applications is difficulty interpreting outcomes This paper focuses on combinatorics topologies along a line segment, an intrinsic geodesic metric, between two phylogenetic trees space and w...