We construct an explicit algorithm of the static-preserving bijection between the rigged configurations and the highest weight paths of the form $(B^{2,1})^{otimes L}$ in the $G_{2}^{(1)}$ adjoint crystals.
For each positive integer $n$, we construct a bijection between the odd partitions and the distinct partitions of $n$ which extends Bressouds bijection between the odd-and-distinct partitions of $n$ and the splitting partitions of $n$. We compare o
ur bijection with the classical bijections of Glaisher and Sylvester, and also with a recent bijection due to Chen, Gao, Ji and Li.
We show that sequences A026737 and A111279 in The On-Line Encyclopedia of Integer Sequences are the same by giving a bijection between two classes of Grand Schroder paths.
We present an algorithm that produces the classification list of smooth Fano d-polytopes for any given d. The input of the algorithm is a single number, namely the positive integer d. The algorithm has been used to classify smooth Fano d-polytopes fo
r d<=7. There are 7622 isomorphism classes of smooth Fano 6-polytopes and 72256 isomorphism classes of smooth Fano 7-polytopes.
We give an approximation algorithm for MaxCut and provide guarantees on the average fraction of edges cut on $d$-regular graphs of girth $geq 2k$. For every $d geq 3$ and $k geq 4$, our approximation guarantees are better than those of all other clas
sical and quantum algorithms known to the authors. Our algorithm constructs an explicit vector solution to the standard semidefinite relaxation of MaxCut and applies hyperplane rounding. It may be viewed as a simplification of the previously best known technique, which approximates Gaussian wave processes on the infinite $d$-regular tree.
We model the optical to far-infrared SEDs of a sample of six type-1 and six type-2 quasars selected in the mid-infrared. The objects in our sample are matched in mid-IR luminosity and selected based on their Spitzer IRAC colors. We obtained new targe
ted Spitzer IRS and MIPS observations and used archival photometry to examine the optical to far-IR SEDs. We investigate whether the observed differences between samples are consistent with orientation-based unification schemes. The type-1 objects show significant emission at 3 micron. They do not show strong PAH emission and have less far-IR emission on average when compared to the type-2 objects. The SEDs of the type-2 objects show a wide assortment of silicate features, ranging from weak emission to deep silicate absorption. Some also show strong PAH features. In comparison, silicate is only seen in emission in the type-1 objects. This is consistent with some of the type-2s being reddened by a foreground screen of cooler dust, perhaps in the host galaxy itself. We investigate the AGN contribution to the far-IR emission and find it to be significant. We also estimate the star formation rate for each of the objects by integrating the modeled far-IR flux and compare this with the SFR found from PAH emission. We find the type-2 quasars have a higher average SFR than the type-1 quasars based on both methods, though this could be due to differences in bolometric luminosities of the objects. While we find pronounced differences between the two types of objects, none of them are inconsistent with orientation-based unification schemes.