Combinatorial Invariants of Rational Polytopes

The first part of this dissertation deals with the equivariant Ehrhart theory of the permutahedron. As a starting point to determining the equivariant Ehrhart theory of the permutahedron, Ardila, Schindler, and I obtain a volume formula for the rational polytopes that are fixed by acting on the permutahedron by a permutation, which generalizes a result of Stanley’s for the volume for the standard permutahedron. Building from the aforementioned work, Ardila, Supina, and I determine the equivariant Ehrhart theory of the permutahedron, thereby resolving an open problem posed by Stapledon. We provide combinatorial descriptions of the Ehrhart quasipolynomial and Ehrhart series of the fixed polytopes of the permutahedron. Additionally, we answer questions regarding the polynomiality of the equivariant analogue of the h*-polynomial. The second part of this dissertation deals with decompositions of the h*-polynomial for rational polytopes. An open problem in Ehrhart theory is to classify all Ehrhart quasipolynomials. Toward this classification problem, one may ask for necessary in- equalities among the coefficients of an h*-polynomial. Beck, Braun, and I contribute such inequalities when P is a rational polytope. Additionally, we provide two decompositions of the h*-polynomial for rational polytopes, thereby generalizing results of Betke and McMullen and Stapledon. We use our rational Betke–McMullen formula to provide a novel proof of Stanley’s Monotonicity Theorem for rational polytopes

A Brief Survey on Lattice Zonotopes

Zonotopes are a rich and fascinating family of polytopes, with connections to many areas of mathematics. In this article we provide a brief survey of classical and recent results related to lattice zonotopes. Our emphasis is on connections to combinatorics, both in the sense of enumeration (e.g. Ehrhart theory) and combinatorial structures (e.g. graphs and permutations)

Elemental abundance differences in the 16 Cygni binary system: a signature of gas giant planet formation?

The atmospheric parameters of the components of the 16Cygni binary system, in which the secondary has a gas giant planet detected, are measured accurately using high quality observational data. Abundances relative to solar are obtained for 25 elements with a mean error of 0.023 dex. The fact that 16CygA has about four times more lithium than 16CygB is normal considering the slightly different masses of the stars. The abundance patterns of 16CygA and B, relative to iron, are typical of that observed in most of the so-called solar twin stars, with the exception of the heavy elements (Z>30), which can, however, be explained by Galactic chemical evolution. Differential (A-B) abundances are measured with even higher precision (0.018 dex, on average). We find that 16CygA is more metal-rich than 16CygB by 0.041+/-0.007 dex. On an element-to-element basis, no correlation between the A-B abundance differences and dust condensation temperature (Tc) is detected. Based on these results, we conclude that if the process of planet formation around 16CygB is responsible for the observed abundance pattern, the formation of gas giants produces a constant downwards shift in the photospheric abundance of metals, without a Tc correlation. The latter would be produced by the formation of terrestrial planets instead, as suggested by other recent works on precise elemental abundances. Nevertheless, a scenario consistent with these observations requires the convective envelopes of 1 Msun stars to reach their present-day sizes about three times quicker than predicted by standard stellar evolution models.Comment: ApJ, in pres

Sedimentary heterogeneity and petrophysical characterization of Barremian tsunami and barrier island/inlet deposits: The Aliaga outcrop as a reservoir analogue (Galve sub-basin, eastern Spain)

The present study examined two sandstone deposits in the Aliaga outcrop as a reservoir analogue over a distance of 200-m-long and attempted to establish a correlation between sand facies and the petrophysical properties of the sandstones in order to investigate the reservoir heterogeneity. The Aliaga reservoir analogue represents the upper part of Camarillas Fm., deposited during the Barremian synrift phase of the Galve sub-basin (Iberian Basin, Spain). It is characterized by a transitional sedimentary interval from sandy-dominant deposits to carbonate-dominant deposits, which were deposited under the same palaeoenvironmental conditions (in relation to systems of back-barrier sedimentation). The description of the Aliaga outcrop provided here consists of lithological descriptions of two sandstone deposits: a tsunami and a barrier island/inlet, at both mesoscopic (decimetres to tens of metres) and microscopic scales (millimetres to centimetres). Both deposits recognized at the basin scale were described in terms of sand grain size, sand sorting and cementation; further cores were drilled along outcrop to collect samples for porosity and permeability measurements. Both sandstone reservoirs are the result of different sedimentary processes that determined facies characteristics, as the different petrophysical properties observed in these deposits. Consequently, the sedimentary process controls the heterogeneity of the sandstones facies and thus, the sand heterogeneity controls the distribution of the petrophysical properties. The classification of sand facies in terms of sand sorting seems to be more appropriate for describing sand heterogeneity; accordingly, petrophysical parameters in both deposits were also influenced by sand sorting. The sand facies and petrophysics heterogeneity of the described deposits can be hierarchically ordered. First-order heterogeneity is related to the basin scale, second-order heterogeneity is related to genesis and the conditions of sediment deposition, and third-order heterogeneity is related to synsedimentary faults and/or post-sedimentation events.This research is a contribution to the project: Análisis de Cuencas Sedimentarias Continentales, of the Gobierno de Aragón, the Análisis de Cuencas Sedimentarias Group of the UCM-CAM, and the projects CGL2011-23717 (Ministerio de Ciencia e Innovación of the Spanish Government and FEDER) and UZ2015-CIE-10 (University of Zaragoza). Additional financial support was provided by a Cnpq (Conselho Nacional de Pesquisa e Desenvolvimento Tecnologico, Brasil) Research Grant (200147/2011-0) and an IAS Schema 1st session 2014 Postgraduate Grant to F. Veloso.Peer reviewe

Generalized parking function polytopes

A classical parking function of length $n$ is a list of positive integers $(a_1, a_2, \ldots, a_n)$ whose nondecreasing rearrangement $b_1 \leq b_2 \leq \cdots \leq b_n$ satisfies $b_i \leq i$. The convex hull of all parking functions of length $n$ is an $n$-dimensional polytope in $\mathbb{R}^n$, which we refer to as the classical parking function polytope. Its geometric properties have been explored in (Amanbayeva and Wang 2022) in response to a question posed in (Stanley 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of $\mathbf{x}$-parking functions for $\mathbf{x}=(a,b,\dots,b)$, which we refer to as $\mathbf{x}$-parking function polytopes. We explore connections between these $\mathbf{x}$-parking function polytopes, the Pitman-Stanley polytope, and the partial permutahedra of (Heuer and Striker 2022). In particular, we establish a closed-form expression for the volume of $\mathbf{x}$-parking function polytopes. This allows us to answer a conjecture of (Behrend et al. 2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary.Comment: 29 pages, 3 figures, comments welcome

New Indicators for AGN Power: The Correlation Between [O IV] lambda 25.89 micron and Hard X-ray Luminosity for Nearby Seyfert Galaxies

We have studied the relationship between the [O IV] lambda 25.89 micron emission line luminosities, obtained from Spitzer spectra, the X-ray continua in the 2-10 keV band, primarily from ASCA, and the 14-195 keV band obtained with the SWIFT/Burst Alert Telescope (BAT), for a sample of nearby (z < 0.08) Seyfert galaxies. For comparison, we have examined the relationship between the [O III] 5007, the 2-10 keV and the 14-195 keV luminosities for the same set of objects. We find that both the [O IV] and [O III] luminosities are well-correlated with the BAT luminosities. On the other hand, the [O III] luminosities are better-correlated with 2-10 keV luminosities than are those of [O IV]. When comparing [O IV] and [O III] luminosities for the different types of galaxies, we find that the Seyfert 2's have significantly lower [O III] to [O IV] ratios than the Seyfert 1's. We suggest that this is due to more reddening of the narrow line region (NLR) of the Seyfert 2's. Assuming Galactic dust to gas ratios, the average amount of extra reddening corresponds to a hydrogen column density of ~ few times 10^21 cm^-2, which is a small fraction of the X-ray absorbing columns in the Seyfert 2's. The combined effects of reddening and the X-ray absorption are the probable reason why the [O III] versus 2-10 keV correlation is better than the [O IV] versus 2-10 keV, since the [O IV] emission line is much less affected by extinction. Overall, we find the [O IV] to be an accurate and truly isotropic indicator of the power of the AGN. This suggests that it can be useful in deconvolving the contribution of the AGN and starburst to the spectrum of Compton-thick and/or X-ray weak sources.Comment: Accepted for publication in the Astrophysical Journal. 31 pages, 6 figures, 4 table