Drawing Planar Graphs with a Prescribed Inner Face

Given a plane graph $G$ (i.e., a planar graph with a fixed planar embedding) and a simple cycle $C$ in $G$ whose vertices are mapped to a convex polygon, we consider the question whether this drawing can be extended to a planar straight-line drawing of $G$. We characterize when this is possible in terms of simple necessary conditions, which we prove to be sufficient. This also leads to a linear-time testing algorithm. If a drawing extension exists, it can be computed in the same running time

Random geometric complexes

We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H_k is not monotone when k > 0. In particular for every k > 0 we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. The main technical contribution of the article is in the application of discrete Morse theory in geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete & Computational Geometr

Irreducible triangulations of surfaces with boundary

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was known only for surfaces without boundary (b=0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary

The Effects of Atmospheric Dispersion on High-Resolution Solar Spectroscopy

We investigate the effects of atmospheric dispersion on observations of the Sun at the ever-higher spatial resolutions afforded by increased apertures and improved techniques. The problems induced by atmospheric refraction are particularly significant for solar physics because the Sun is often best observed at low elevations, and the effect of the image displacement is not merely a loss of efficiency, but the mixing of information originating from different points on the solar surface. We calculate the magnitude of the atmospheric dispersion for the Sun during the year and examine the problems produced by this dispersion in both spectrographic and filter observations. We describe an observing technique for scanning spectrograph observations that minimizes the effects of the atmospheric dispersion while maintaining a regular scanning geometry. Such an approach could be useful for the new class of high-resolution solar spectrographs, such as SPINOR, POLIS, TRIPPEL, and ViSP

Can induced gravity isotropize Bianchi I, V, or IX Universes?

We analyze if Bianchi I, V, and IX models in the Induced Gravity (IG) theory can evolve to a Friedmann--Roberson--Walker (FRW) expansion due to the non--minimal coupling of gravity and the scalar field. The analytical results that we found for the Brans-Dicke (BD) theory are now applied to the IG theory which has $\omega \ll 1$ ($\omega$ being the square ratio of the Higgs to Planck mass) in a cosmological era in which the IG--potential is not significant. We find that the isotropization mechanism crucially depends on the value of $\omega$. Its smallness also permits inflationary solutions. For the Bianch V model inflation due to the Higgs potential takes place afterwads, and subsequently the spontaneous symmetry breaking (SSB) ends with an effective FRW evolution. The ordinary tests of successful cosmology are well satisfied.Comment: 24 pages, 5 figures, to be published in Phys. Rev. D1

Domain Wall Spacetimes: Instability of Cosmological Event and Cauchy Horizons

The stability of cosmological event and Cauchy horizons of spacetimes associated with plane symmetric domain walls are studied. It is found that both horizons are not stable against perturbations of null fluids and massless scalar fields; they are turned into curvature singularities. These singularities are light-like and strong in the sense that both the tidal forces and distortions acting on test particles become unbounded when theses singularities are approached.Comment: Latex, 3 figures not included in the text but available upon reques

Oligarchic growth of giant planets

Runaway growth ends when the largest protoplanets dominate the dynamics of the planetesimal disk; the subsequent self-limiting accretion mode is referred to as ``oligarchic growth.'' Here, we begin by expanding on the existing analytic model of the oligarchic growth regime. From this, we derive global estimates of the planet formation rate throughout a protoplanetary disk. We find that a relatively high-mass protoplanetary disk ($\sim$ 10$\times$ minimum-mass) is required to produce giant planet core-sized bodies ($\sim$ 10 M$_{\oplus}$) within the lifetime of the nebular gas (\la 10 million years). However, an implausibly massive disk is needed to produce even an Earth mass at the orbit of Uranus by 10 Myrs. Subsequent accretion without the dissipational effect of gas is even slower and less efficient. In the limit of non-interacting planetesimals, a reasonable-mass disk is unable to produce bodies the size of the Solar System's two outer giant planets at their current locations on {\it any} timescale; if collisional damping of planetesimal random velocities is sufficiently effective, though, it may be possible for a Uranus/Neptune to form in situ in less than the age of the Solar System. We perform numerical simulations of oligarchic growth with gas, and find that protoplanet growth rates agree reasonably well with the analytic model as long as protoplanet masses are well below their estimated final masses. However, accretion stalls earlier than predicted, so that the largest final protoplanet masses are smaller than those given by the model. Thus the oligarchic growth model, in the form developed here, appears to provide an upper limit for the efficiency of giant planet formation.Comment: ScienceDirect already has the final published version here: dx.doi.org/10.1016/S0019-1035(02)00043-

Performance of novel VUV-sensitive Silicon Photo-Multipliers for nEXO

Liquid xenon time projection chambers are promising detectors to search for neutrinoless double beta decay (0$\nu \beta \beta$), due to their response uniformity, monolithic sensitive volume, scalability to large target masses, and suitability for extremely low background operations. The nEXO collaboration has designed a tonne-scale time projection chamber that aims to search for 0$\nu \beta \beta$ of \ce{^{136}Xe} with projected half-life sensitivity of $1.35\times 10^{28}$~yr. To reach this sensitivity, the design goal for nEXO is $\leq$1\% energy resolution at the decay $Q$-value ($2458.07\pm 0.31$~keV). Reaching this resolution requires the efficient collection of both the ionization and scintillation produced in the detector. The nEXO design employs Silicon Photo-Multipliers (SiPMs) to detect the vacuum ultra-violet, 175 nm scintillation light of liquid xenon. This paper reports on the characterization of the newest vacuum ultra-violet sensitive Fondazione Bruno Kessler VUVHD3 SiPMs specifically designed for nEXO, as well as new measurements on new test samples of previously characterised Hamamatsu VUV4 Multi Pixel Photon Counters (MPPCs). Various SiPM and MPPC parameters, such as dark noise, gain, direct crosstalk, correlated avalanches and photon detection efficiency were measured as a function of the applied over voltage and wavelength at liquid xenon temperature (163~K). The results from this study are used to provide updated estimates of the achievable energy resolution at the decay $Q$-value for the nEXO design

What is the Oxygen Isotope Composition of Venus? The Scientific Case for Sample Return from Earth’s “Sister” Planet

Venus is Earth’s closest planetary neighbour and both bodies are of similar size and mass. As a consequence, Venus is often described as Earth’s sister planet. But the two worlds have followed very different evolutionary paths, with Earth having benign surface conditions, whereas Venus has a surface temperature of 464 °C and a surface pressure of 92 bar. These inhospitable surface conditions may partially explain why there has been such a dearth of space missions to Venus in recent years.The oxygen isotope composition of Venus is currently unknown. However, this single measurement (Δ17O) would have first order implications for our understanding of how large terrestrial planets are built. Recent isotopic studies indicate that the Solar System is bimodal in composition, divided into a carbonaceous chondrite (CC) group and a non-carbonaceous (NC) group. The CC group probably originated in the outer Solar System and the NC group in the inner Solar System. Venus comprises 41% by mass of the inner Solar System compared to 50% for Earth and only 5% for Mars. Models for building large terrestrial planets, such as Earth and Venus, would be significantly improved by a determination of the Δ17O composition of a returned sample from Venus. This measurement would help constrain the extent of early inner Solar System isotopic homogenisation and help to identify whether the feeding zones of the terrestrial planets were narrow or wide.Determining the Δ17O composition of Venus would also have significant implications for our understanding of how the Moon formed. Recent lunar formation models invoke a high energy impact between the proto-Earth and an inner Solar System-derived impactor body, Theia. The close isotopic similarity between the Earth and Moon is explained by these models as being a consequence of high-temperature, post-impact mixing. However, if Earth and Venus proved to be isotopic clones with respect to Δ17O, this would favour the classic, lower energy, giant impact scenario.We review the surface geology of Venus with the aim of identifying potential terrains that could be targeted by a robotic sample return mission. While the potentially ancient tessera terrains would be of great scientific interest, the need to minimise the influence of venusian weathering favours the sampling of young basaltic plains. In terms of a nominal sample mass, 10 g would be sufficient to undertake a full range of geochemical, isotopic and dating studies. However, it is important that additional material is collected as a legacy sample. As a consequence, a returned sample mass of at least 100 g should be recovered.Two scenarios for robotic sample return missions from Venus are presented, based on previous mission proposals. The most cost effective approach involves a “Grab and Go” strategy, either using a lander and separate orbiter, or possibly just a stand-alone lander. Sample return could also be achieved as part of a more ambitious, extended mission to study the venusian atmosphere. In both scenarios it is critical to obtain a surface atmospheric sample to define the extent of atmosphere-lithosphere oxygen isotopic disequilibrium. Surface sampling would be carried out by multiple techniques (drill, scoop, “vacuum-cleaner” device) to ensure success. Surface operations would take no longer than one hour.Analysis of returned samples would provide a firm basis for assessing similarities and differences between the evolution of Venus, Earth, Mars and smaller bodies such as Vesta. The Solar System provides an important case study in how two almost identical bodies, Earth and Venus, could have had such a divergent evolution. Finally, Venus, with its runaway greenhouse atmosphere, may provide data relevant to the understanding of similar less extreme processes on Earth. Venus is Earth’s planetary twin and deserves to be better studied and understood. In a wider context, analysis of returned samples from Venus would provide data relevant to the study of exoplanetary systems