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Thomas Beck

Publications and source records attributed to Thomas Beck.

36 records · Page 2Linked to original sources
arXiv2022

A pair of Sub-Neptunes transiting the bright K-dwarf TOI-1064 characterised with CHEOPS

We report the discovery and characterisation of a pair of sub-Neptunes transiting the bright K-dwarf TOI-1064 (TIC 79748331), initially detected in TESS photometry. To characterise the system, we performed and retrieved CHEOPS, TESS, and ground-based photometry, HARPS high-resolution spectroscopy, and Gemini speckle imaging. We characterise the host star and determine $T_{\rm eff, \star}=4734\pm67$ K, $R_{\star}=0.726\pm0.007$ $R_{\odot}$, and $M_{\star}=0.748\pm0.032$ $M_{\odot}$. We present a novel detrending method based on PSF shape-change modelling and demonstrate its suitability to correct flux variations in CHEOPS data. We confirm the planetary nature of both bodies and find that TOI-1064 b has an orbital period of $P_{\rm b}=6.44387\pm0.00003$ d, a radius of $R_{\rm b}=2.59\pm0.04$ $R_{\oplus}$, and a mass of $M_{\rm b}=13.5_{-1.8}^{+1.7}$ $M_{\oplus}$, whilst TOI-1064 c has an orbital period of $P_{\rm c}=12.22657^{+0.00005}_{-0.00004}$ d, a radius of $R_{\rm c}=2.65\pm0.04$ $R_{\oplus}$, and a 3$σ$ upper mass limit of 8.5 ${\rm M_{\oplus}}$. From the high-precision photometry we obtain radius uncertainties of $\sim$1.6%, allowing us to conduct internal structure and atmospheric escape modelling. TOI-1064 b is one of the densest, well-characterised sub-Neptunes, with a tenuous atmosphere that can be explained by the loss of a primordial envelope following migration through the protoplanetary disc. It is likely that TOI-1064 c has an extended atmosphere due to the tentative low density, however further RVs are needed to confirm this scenario and the similar radii, different masses nature of this system. The high-precision data and modelling of TOI-1064 b are important for planets in this region of mass-radius space, and it allows us to identify a trend in bulk density-stellar metallicity for massive sub-Neptunes that may hint at the formation of this population of planets.

astro-ph.EP↗
arXiv2021

Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs

We prove quantitative norm bounds for a family of operators involving impedance boundary conditions on convex, polygonal domains. A robust numerical construction of Helmholtz scattering solutions in variable media via the Dirichlet-to-Neumann operator involves a decomposition of the domain into a sequence of rectangles of varying scales and constructing impedance-to-impedance boundary operators on each subdomain. Our estimates in particular ensure the invertibility, with quantitative bounds in the frequency, of the merge operators required to reconstruct the original Dirichlet-to-Neumann operator in terms of these impedance-to-impedance operators of the sub-domains. A key step in our proof is to obtain Neumann and Dirichlet boundary trace estimates on solutions of the impedance problem, which are of independent interest. In addition to the variable media setting, we also construct bounds for similar merge operators in the obstacle scattering problem.

math.AP↗
arXiv2021

The Friedland-Hayman inequality and Caffarelli's contraction theorem

The Friedland-Hayman inequality is a sharp inequality concerning the growth rates of homogeneous, harmonic functions with Dirichlet boundary conditions on complementary cones dividing Euclidean space into two parts. In this paper, we prove an analogous inequality in which one divides a convex cone into two parts, placing Neumann conditions on the boundary of the convex cone, and Dirichlet conditions on the interface. This analogous inequality was already proved by us jointly with Sarah Raynor. Here we present a new proof that permits us to characterize the case of equality. In keeping with the two-phase free boundary theory introduced by Alt, Caffarelli, and Friedman, such an improvement can be expected to yield further regularity in free boundary problems.

math.AP↗
arXiv2020

The CHEOPS mission

The CHaracterising ExOPlanet Satellite (CHEOPS) was selected in 2012, as the first small mission in the ESA Science Programme and successfully launched in December 2019. CHEOPS is a partnership between ESA and Switzerland with important contributions by ten additional ESA Member States. CHEOPS is the first mission dedicated to search for transits of exoplanets using ultrahigh precision photometry on bright stars already known to host planets. As a follow-up mission, CHEOPS is mainly dedicated to improving, whenever possible, existing radii measurements or provide first accurate measurements for a subset of those planets for which the mass has already been estimated from ground-based spectroscopic surveys and to following phase curves. CHEOPS will provide prime targets for future spectroscopic atmospheric characterisation. Requirements on the photometric precision and stability have been derived for stars with magnitudes ranging from 6 to 12 in the V band. In particular, CHEOPS shall be able to detect Earth-size planets transiting G5 dwarf stars in the magnitude range between 6 and 9 by achieving a photometric precision of 20 ppm in 6 hours of integration. For K stars in the magnitude range between 9 and 12, CHEOPS shall be able to detect transiting Neptune-size planets achieving a photometric precision of 85 ppm in 3 hours of integration. This is achieved by using a single, frame-transfer, back-illuminated CCD detector at the focal plane assembly of a 33.5 cm diameter telescope. The 280 kg spacecraft has a pointing accuracy of about 1 arcsec rms and orbits on a sun-synchronous dusk-dawn orbit at 700 km altitude. The nominal mission lifetime is 3.5 years. During this period, 20% of the observing time is available to the community through a yearly call and a discretionary time programme managed by ESA.

astro-ph.IM↗
arXiv2020

Limiting Eigenfunctions of Sturm-Liouville operators Subject to a Spectral Flow

We examine the spectrum of a family of Sturm--Liouville operators with regularly spaced delta function potentials parametrized by increasing strength. The limiting behavior of the eigenvalues under this spectral flow was described in a previor result of the last two authors with Berkolaiko, where it was used to study the nodal deficiency of Laplacian eigenfunctions. Here we consider the eigenfunctions of these operators. In particular, we give explicit formulas for the limiting eigenfunctions, and also characterize the eigenfunctions and eigenvalues for all values for the spectral flow parameter (not just in the limit). We also develop spectrally accurate numerical tools for comparison and visualization.

math.SP↗
arXiv2020

Two-phase free boundary problems in convex domains

We study the regularity of minimizers of a two-phase free boundary problem. For a class of n-dimensional convex domains, we establish the Lipschitz continuity of the minimizer up to the fixed boundary under Neumann boundary conditions. Our proof uses an almost monotonicity formula for the Alt-Caffarelli-Friedman functional restricted to the convex domain. This requires a variant of the classical Friedland-Hayman inequality for geodesically convex subsets of the sphere with Neumann boundary conditions. To apply this inequality, in addition to convexity, we require a Dini condition governing the rate at which the fixed boundary converges to its limit cone at each boundary point.

math.AP↗
arXiv2019

Localization of the first eigenfunction of a convex domain

We study the first Dirichlet eigenfunction of the Laplacian in a $n$-dimensional convex domain. For domains of a fixed inner radius, estimates of Chiti imply that the ratio of the $L^2$-norm and $L^{\infty}$-norm of the eigenfunction is minimized when the domain is a ball. However, when the eccentricity of the domain is large the eigenfunction should spread out at a certain scale and this ratio should increase. We make this precise by obtaining a lower bound on the $L^2$-norm of the eigenfunction and show that the eigenfunction cannot localize to too small a subset of the domain. As a consequence, we settle a conjecture of van den Berg, in the general $n$-dimensional case. The main feature of the proof is to obtain sufficiently sharp estimates on the first eigenvalue in order to estimate the first derivatives of the eigenfunction.

math.AP↗
arXiv2019

Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions

Let $Ω\subset \mathbb{R}^n$ be a convex domain and let $f:Ω\rightarrow \mathbb{R}$ be a positive, subharmonic function (i.e. $Δf \geq 0$). Then $$ \frac{1}{|Ω|} \int_Ω{f dx} \leq \frac{c_n}{ |\partial Ω| } \int_{\partial Ω}{ f dσ},$$ where $c_n \leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n \geq n-1$. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other $ Ω_2 \subset Ω_1 \subset \mathbb{R}^n$: $$ \frac{|\partial Ω_1|}{|Ω_1|} \frac{| Ω_2|}{|\partial Ω_2|} \leq n.$$

math.CA↗
arXiv2019

Nodal line estimates for the second Dirichlet eigenfunction

We study the nodal curves of low energy Dirichlet eigenfunctions in generalized curvilinear quadrilaterals. The techniques can be seen as a generalization of the tools developed by Grieser-Jerison in a series of works on convex planar domains and rectangles with one curved edge and a large aspect ratio. Here, we study the structure of the nodal curve in greater detail, in that we find precise bounds on its curvature, with uniform estimates up to the two points where it meets the domain at right angles, and show that many of our results hold for relatively small aspect ratios of the side lengths. We also discuss applications of our results to Courant-sharp eigenfunctions and spectral partitioning.

math.AP↗
arXiv2018

The torsion function of convex domains of high eccentricity

The torsion function of a convex planar domain has convex level sets, but explicit formulae are known only for rectangles and ellipses. Here we study the torsion function on convex planar domains of high eccentricity. We obtain an approximation for the torsion function by viewing the domain as a perturbation of a rectangle in order to define an approximate Green's function for the Laplacian. For a class of convex domains we use this approximation to establish sharp bounds on the Hessian and the infinitesimal shape of the level sets around its maximum. We also use these results to compare the behaviour of the torsion function and the first eigenfunction of the Dirichlet Laplacian around their respective maxima.

math.AP↗
arXiv2017

Uniform level set estimates for ground state eigenfunctions

We study the behaviour of the first eigenfunction of the Dirichlet Laplacian on a planar convex domain near its maximum. We show that the eccentricity and orientation of the superlevel sets of the eigenfunction stabilise as they approach the maximum, uniformly with respect to the eccentricity of the domain itself. This is achieved by obtaining quantitatively sharp second derivative estimates, which are consistent with the shape of the superlevel sets. In particular, we prove that the eigenfunction is concave (rather than merely log concave) in an entire superlevel set near its maximum. By estimating the mixed second and third derivatives partial derivatives of the eigenfunction, we also determine the rate at which a degree 2 Taylor polynomial approximates the eigenfunction itself.

math.AP↗
arXiv2017

Level Spacings and Nodal Sets at Infinity for Radial Perturbations of the Harmonic Oscillator

We study properties of the nodal sets of high frequency eigenfunctions and quasimodes for radial perturbations of the Harmonic Oscillator. In particular, we consider nodal sets on spheres of large radius (in the classically forbidden region) for quasimodes with energies lying in intervals around a fixed energy $E$. For well chosen intervals we show that these nodal sets exhibit quantitatively different behavior compared to those of the unperturbed Harmonic Oscillator. These energy intervals are defined via a careful analysis of the eigenvalue spacings for the perturbed operator, based on analytic perturbation theory and linearization formulas for Laguerre polynomials.

math-ph↗
arXiv2016

Nodal Sets of Smooth Functions with Finite Vanishing Order and p-Sweepouts

We show that on a compact Riemmanian manifold $(M,g)$, nodal sets of linear combinations of any $p+1$ smooth functions form an admissible $p-$sweepout provided these linear combinations have uniformly bounded vanishing order. This applies in particular to finite linear combinations of Laplace eigenfunctions. As a result, we obtain a new proof of the Gromov, Guth, Marques--Neves upper bounds on the min-max $p$-widths of $M.$ We also prove that close to a point at which a smooth function on $\mathbb{R}^{n+1}$ vanishes to order $k$, its nodal set is contained in the union of $k$ $W^{1,p}$ graphs for some $p > 1$. This implies that the nodal set is locally countably $n$-rectifiable and has locally finite $\mathcal{H}^n$ measure, facts which also follow from a previous result of Bär. Finally, we prove the continuity of the Hausdorff measure of nodal sets under heat flow.

math.AP↗
arXiv2015

On Global Solutions of a Zakharov type System

We consider a class of wave-Schroedinger systems with a Zakharov-Schulman type coupling. This class of systems is indexed by a parameter gamma which measures the strength of the null form in the nonlinearity of the wave equation. The case gamma = 1 corresponds to the well-known Zakharov system, while the case gamma = -1 corresponds to the Yukawa system. Here we show that sufficiently smooth and localized Cauchy data lead to pointwise decaying global solutions which scatter, for any gamma in (0,1].

math.AP↗
arXiv2014

The Shape of the Level Sets of the First Eigenfunction of a Class of Two Dimensional Schrödinger Operators

We study the first Dirichlet eigenfunction of a class of Schrödinger operators with a convex potential V on a domain $Ω$. We find two length scales $L_1$ and $L_2$, and an orientation of the domain $Ω$, which determine the shape of the level sets of the eigenfunction. As an intermediate step, we also establish bounds on the first eigenvalue in terms of the first eigenvalue of an associated ordinary differential operator.

math.AP↗
arXiv2009

Breakdown Current Density of Graphene Nano Ribbons

Graphene nanoribbons (GNRs) with widths down to 16 nm have been characterized for their current-carrying capacity. It is found that GNRs exhibit an impressive breakdown current density, on the order of 10^8 A/cm2. The breakdown current density is found to have a reciprocal relationship to GNR resistivity and the data fit points to Joule heating as the likely mechanism of breakdown. The superior current-carrying capacity of GNRs will be valuable for their application in on-chip electrical interconnects. The thermal conductivity of sub-20 nm graphene ribbons is found to be more than 1000 W/m-K.

cond-mat.mes-hall↗
arXiv2009

Resistivity of Graphene Nanoribbon Interconnects

Graphene nanoribbon interconnects are fabricated, and the extracted resistivity is compared to that of Cu. It is found that the average resistivity at a given line-width (18nm<W<52nm) is about 3X that of a Cu wire, whereas the best GNR has a resistivity comparable to that of Cu. The conductivity is found to be limited by impurity scattering as well as LER scattering; as a result, the best reported GNR resistivity is 3X the limit imposed by substrate phonon scattering. This study reveals that even moderate-quality graphene nanowires have the potential to outperform Cu for use as on-chip interconnects.

cond-mat.mes-hall↗