The Inverse Eigenvalue Problem of a Graph

Inverse eigenvalue problems appear in various contexts throughout mathematics and engineering, and refer to determining all possible lists of eigenvalues (spectra) for matrices fitting some description. The inverse eigenvalue problem of a graph refers to determining the possible spectra of real symmetric matrices whose pattern of nonzero off-diagonal entries is described by the edges of a given graph (precise definitions of this and other terms are given in the next paragraph). This problem and related variants have been of interest for many years and were originally approached through the study of ordered multiplicity lists.This report resulted from the Banff International Research Station Focused Research Groups and is published as Barrett, Wayne, Steve Butler, Shaun Fallat, H. Tracy Hall, Leslie Hogben, Jephian CH Lin, Bryan Shader, and Michael Young. "The inverse eigenvalue problem of a graph." Banff International Research Station: The Inverse Eigenvalue Problem of a Graph, 2016. Posted with permission.</p

The Enhanced Principal Rank Characteristic Sequence

The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric n×n matrix is a sequence ℓ1ℓ2⋯ℓn where ℓk is A, S, or N according as all, some, or none of its principal minors of order k are nonzero. Such sequences give more information than the (0,1) pr-sequences previously studied (where basically the kth entry is 0 or 1 according as none or at least one of its principal minors of order k is nonzero). Various techniques including the Schur complement are introduced to establish that certain subsequences such as NAN are forbidden in epr-sequences over fields of characteristic not two. Using probabilistic methods over fields of characteristic zero, it is shown that any sequence of As and Ss ending in A is attainable, and any sequence of As and Ss followed by one or more Ns is attainable; additional families of attainable epr-sequences are constructed explicitly by other methods. For real symmetric matrices of orders 2, 3, 4, and 5, all attainable epr-sequences are listed with justifications

Radioactive Probes of the Supernova-Contaminated Solar Nebula: Evidence that the Sun was Born in a Cluster

We construct a simple model for radioisotopic enrichment of the protosolar nebula by injection from a nearby supernova, based on the inverse square law for ejecta dispersion. We find that the presolar radioisotopes abundances (i.e., in solar masses) demand a nearby supernova: its distance can be no larger than 66 times the size of the protosolar nebula, at a 90% confidence level, assuming 1 solar mass of protosolar material. The relevant size of the nebula depends on its state of evolution at the time of radioactivity injection. In one scenario, a collection of low-mass stars, including our sun, formed in a group or cluster with an intermediate- to high-mass star that ended its life as a supernova while our sun was still a protostar, a starless core, or perhaps a diffuse cloud. Using recent observations of protostars to estimate the size of the protosolar nebula constrains the distance of the supernova at 0.02 to 1.6 pc. The supernova distance limit is consistent with the scales of low-mass stars formation around one or more massive stars, but it is closer than expected were the sun formed in an isolated, solitary state. Consequently, if any presolar radioactivities originated via supernova injection, we must conclude that our sun was a member of such a group or cluster that has since dispersed, and thus that solar system formation should be understood in this context. In addition, we show that the timescale from explosion to the creation of small bodies was on the order of 1.8 Myr (formal 90% confidence range of 0 to 2.2 Myr), and thus the temporal choreography from supernova ejecta to meteorites is important. Finally, we can not distinguish between progenitor masses from 15 to 25 solar masses in the nucleosynthesis models; however, the 20 solar mass model is somewhat preferred.Comment: ApJ accepted, 19 pages, 3 figure

The mass and density of the dwarf planet (225088) 2007 OR10

The satellite of (225088) 2007 OR10 was discovered on archival Hubble Space Telescope images and along with new observations with the WFC3 camera in late 2017 we have been able to determine the orbit. The orbit's notable eccentricity, e$\approx$0.3, may be a consequence of an intrinsically eccentric orbit and slow tidal evolution, but may also be caused by the Kozai mechanism. Dynamical considerations also suggest that the moon is small, D$_{eff}$ $<$ 100 km. Based on the newly determined system mass of 1.75x10$^{21}$ kg, 2007 OR10 is the fifth most massive dwarf planet after Eris, Pluto, Haumea and Makemake. The newly determined orbit has also been considered as an additional option in our radiometric analysis, provided that the moon orbits in the equatorial plane of the primary. Assuming a spherical shape for the primary this approach provides a size of 1230$\pm$50 km, with a slight dependence on the satellite orbit orientation and primary rotation rate chosen, and a bulk density of 1.75$\pm$0.07 g cm$^{-3}$ for the primary. A previous size estimate that assumed an equator-on configuration (1535$^{+75}_{-225}$ km) would provide a density of 0.92$^{+0.46}_{-0.14}$ g cm$^{-3}$, unexpectedly low for a 1000 km-sized dwarf planet.Comment: Accepted for publication in Icaru

Genome-Wide Association Study Identifies Loci for Liver Enzyme Concentrations in Mexican Americans: The GUARDIAN Consortium.

ObjectivePopulations of Mexican American ancestry are at an increased risk for nonalcoholic fatty liver disease. The objective of this study was to determine whether loci in known and novel genes were associated with variation in aspartate aminotransferase (AST) (n = 3,644), alanine aminotransferase (ALT) (n = 3,595), and gamma-glutamyl transferase (GGT) (n = 1,577) levels by conducting the first genome-wide association study (GWAS) of liver enzymes, which commonly measure liver function, in individuals of Mexican American ancestry.MethodsLevels of AST, ALT, and GGT were determined by enzymatic colorimetric assays. A multi-cohort GWAS of individuals of Mexican American ancestry was performed. Single-nucleotide polymorphisms (SNP) were tested for association with liver outcomes by multivariable linear regression using an additive genetic model. Association analyses were conducted separately in each cohort, followed by a nonparametric meta-analysis.ResultsIn the PNPLA3 gene, rs4823173 (P = 3.44 × 10-10 ), rs2896019 (P = 7.29 × 10-9 ), and rs2281135 (P = 8.73 × 10-9 ) were significantly associated with AST levels. Although not genome-wide significant, these same SNPs were the top hits for ALT (P = 7.12 × 10-8 , P = 1.98 × 10-7 , and P = 1.81 × 10-7 , respectively). The strong correlation (r2  = 1.0) for these SNPs indicated a single hit in the PNPLA3 gene. No genome-wide significant associations were found for GGT.ConclusionsPNPLA3, a locus previously identified with ALT, AST, and nonalcoholic fatty liver disease in European and Japanese GWAS, is also associated with liver enzymes in populations of Mexican American ancestry

The principal rank characteristic sequence over various fields

Given an n x n matrix, its principal rank characteristic sequence is a sequence of length n+1 of 0s and 1s where, for k = 0, 1, . . . , n, a 1 in the kth position indicates the existence of a principal submatrix of rank k and a 0 indicates the absence of such a submatrix. The principal rank characteristic sequences for symmetric matrices over various fields are investigated, with all such attainable sequences determined for all n over any field with characteristic 2. A complete list of attainable sequences for real symmetric matrices of order 7 is reported