How Well Do We Know Europa’s Topography? An Evaluation of the Variability in Digital Terrain Models of Europa

Jupiter’s moon Europa harbors one of the most likely environments for extant extraterrestrial life. Determining whether Europa is truly habitable requires understanding the structure and thickness of its ice shell, including the existence of perched water or brines. Stereo-derived topography from images acquired by NASA Galileo’s Solid State Imager (SSI) of Europa are often used as a constraint on ice shell structure and heat flow, but the uncertainty in such topography has, to date, not been rigorously assessed. To evaluate the current uncertainty in Europa’s topography we generated and compared digital terrain models (DTMs) of Europa from SSI images using both the open-source Ames Stereo Pipeline (ASP) software and the commercial SOCET SET® software. After first describing the criteria for assessing stereo quality in detail, we qualitatively and quantitatively describe both the horizontal resolution and vertical precision of the DTMs. We find that the horizontal resolution of the SOCET SET® DTMs is typically 8–11× the root mean square (RMS) pixel scale of the images, whereas the resolution of the ASP DTMs is 9–13× the maximum pixel scale of the images. We calculate the RMS difference between the ASP and SOCET SET® DTMs as a proxy for the expected vertical precision (EP), which is a function of the matching accuracy and stereo geometry. We consistently find that the matching accuracy is ~0.5 pixels, which is larger than well-established “rules of thumb” that state that the matching accuracy is 0.2–0.3 pixels. The true EP is therefore ~1.7× larger than might otherwise be assumed. In most cases, DTM errors are approximately normally distributed, and errors that are several times the derived EP occur as expected. However, in two DTMs, larger errors (differences) occur and correlate with real topography. These differences primarily result from manual editing of the SOCET SET® DTMs. The product of the DTM error and the resolution is typically 4–8 pixel2 if calculated using the RMS image scale for SOCET SET® DTMs and the maximum images scale for the ASP DTMs, which is consistent with recent work using martian data sets and suggests that the relationship applies more broadly. We evaluate how ASP parameters affect DTM quality and find that using a smaller subpixel refinement kernel results in DTMs with smaller (better) resolution but, in some cases, larger gaps, which are sometimes reduced by increasing the size of the correlation kernel. We conclude that users of ASP should always systematically evaluate the choice of parameters for a given dataset.

Partial isometries, duality, and determinantal point processes

A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures [Formula: see text] on a space [Formula: see text] with measure [Formula: see text], whose correlation functions are all given by determinants specified by an integral kernel [Formula: see text] called the correlation kernel. We consider a pair of Hilbert spaces, [Formula: see text], which are assumed to be realized as [Formula: see text]-spaces, [Formula: see text], [Formula: see text], and introduce a bounded linear operator [Formula: see text] and its adjoint [Formula: see text]. We show that if [Formula: see text] is a partial isometry of locally Hilbert–Schmidt class, then we have a unique DPP [Formula: see text] associated with [Formula: see text]. In addition, if [Formula: see text] is also of locally Hilbert–Schmidt class, then we have a unique pair of DPPs, [Formula: see text], [Formula: see text]. We also give a practical framework which makes [Formula: see text] and [Formula: see text] satisfy the above conditions. Our framework to construct pairs of DPPs implies useful duality relations between DPPs making pairs. For a correlation kernel of a given DPP our formula can provide plural different expressions, which reveal different aspects of the DPP. In order to demonstrate these advantages of our framework as well as to show that the class of DPPs obtained by this method is large enough to study universal structures in a variety of DPPs, we report plenty of examples of DPPs in one-, two- and higher-dimensional spaces [Formula: see text], where several types of weak convergence from finite DPPs to infinite DPPs are given. One-parameter ([Formula: see text]) series of infinite DPPs on [Formula: see text] and [Formula: see text] are discussed, which we call the Euclidean and the Heisenberg families of DPPs, respectively, following the terminologies of Zelditch.

Global and local scaling limits for the β = 2 Stieltjes–Wigert random matrix ensemble

The eigenvalue probability density function (PDF) for the Gaussian unitary ensemble has a well-known analogy with the Boltzmann factor for a classical log-gas with pair potential [Formula: see text], confined by a one-body harmonic potential. A generalization is to replace the pair potential by [Formula: see text]. The resulting PDF first appeared in the statistical physics literature in relation to non-intersecting Brownian walkers, equally spaced at time [Formula: see text], and subsequently in the study of quantum many-body systems of the Calogero–Sutherland type, and also in Chern–Simons field theory. It is an example of a determinantal point process with correlation kernel based on the Stieltjes–Wigert polynomials. We take up the problem of determining the moments of this ensemble, and find an exact expression in terms of a particular little [Formula: see text]-Jacobi polynomial. From their large [Formula: see text] form, the global density can be computed. Previous work has evaluated the edge scaling limit of the correlation kernel in terms of the Ramanujan ([Formula: see text]-Airy) function. We show how in a particular [Formula: see text] scaling limit, this reduces to the Airy kernel.

Matrix orthogonality in the plane versus scalar orthogonality in a Riemann surface

We consider a non-Hermitian matrix orthogonality on a contour in the complex plane. Given a diagonalizable and rational matrix valued weight, we show that the Christoffel–Darboux (CD) kernel, which is built in terms of matrix orthogonal polynomials, is equivalent to a scalar valued reproducing kernel of meromorphic functions in a Riemann surface. If this Riemann surface has genus $0$, then the matrix valued CD kernel is equivalent to a scalar reproducing kernel of polynomials in the plane. Interestingly, this scalar reproducing kernel is not necessarily a scalar CD kernel. As an application of our result, we show that the correlation kernel of certain doubly periodic lozenge tiling models admits a double contour integral representation involving only a scalar CD kernel. This simplifies a formula of Duits and Kuijlaars.

Critical edge behavior in the singularly perturbed Pollaczek–Jacobi type unitary ensemble

We investigate the orthogonal polynomials associated with a singularly perturbed Pollaczek–Jacobi type weight [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Based on our observation, we find that this weight includes the symmetric constraint [Formula: see text]. Our main results obtained here include two aspects: (1) Strong asymptotics: we deduce strong asymptotics of monic orthogonal polynomials with respect to the above weight function in different regions in the complex plane when the polynomial degree [Formula: see text] goes to infinity. Because of the effect of [Formula: see text] for varying [Formula: see text], the asymptotic behavior in a neighborhood of point [Formula: see text] is described in terms of the Airy function as [Formula: see text], but the Bessel function as [Formula: see text]. Due to symmetry, the similar local asymptotic behavior near the singular point [Formula: see text] can be derived. (2) Limiting eigenvalue correlation kernels: We calculate the limit of the eigenvalue correlation kernel of the corresponding unitary random matrix ensemble in the bulk of the spectrum described by the sine kernel, and at both sides of hard edge, expressed as a Painlevé III kernel. Our analysis is based on the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems.

Interpretations of ground-state symmetry breaking and strong correlation in wavefunction and density functional theories

Strong correlations within a symmetry-unbroken ground-state wavefunction can show up in approximate density functional theory as symmetry-broken spin densities or total densities, which are sometimes observable. They can arise from soft modes of fluctuations (sometimes collective excitations) such as spin-density or charge-density waves at nonzero wavevector. In this sense, an approximate density functional for exchange and correlation that breaks symmetry can be more revealing (albeit less accurate) than an exact functional that does not. The examples discussed here include the stretched H2 molecule, antiferromagnetic solids, and the static charge-density wave/Wigner crystal phase of a low-density jellium. Time-dependent density functional theory is used to show quantitatively that the static charge-density wave is a soft plasmon. More precisely, the frequency of a related density fluctuation drops to zero, as found from the frequency moments of the spectral function, calculated from a recent constraint-based wavevector- and frequency-dependent jellium exchange-correlation kernel.

Determinantal Structures in Space-Inhomogeneous Dynamics on Interlacing Arrays

We introduce a space-inhomogeneous generalization of the dynamics on interlacing arrays considered by Borodin and Ferrari (Commun Math Phys 325:603–684, 2014). We show that for a certain class of initial conditions the point process associated with the dynamics has determinantal correlation functions, and we calculate explicitly, in the form of a double contour integral, the correlation kernel for one of the most classical initial conditions, the densely packed. En route to proving this, we obtain some results of independent interest on non-intersecting general pure-birth chains, that generalize the Charlier process, the discrete analogue of Dyson’s Brownian motion. Finally, these dynamics provide a coupling between the inhomogeneous versions of the TAZRP and PushTASEP particle systems which appear as projections on the left and right edges of the array, respectively.

Design of auxiliary systems for spectroscopy

In this contribution, we advocate the possibility of designing auxiliary systems with effective potentials or kernels that target only the specific spectral properties of interest and are simpler than the self-energy of many-body perturbation theory or the exchange–correlation kernel of time-dependent density-functional theory.

Evaluating the simulation of radiation dose reduction in a digital breast tomosynthesis system featuring an amorphous silicon (a-Si) detector

The validation of many dose optimization methods in x-ray imaging requires clinical images from a range of signal-to-ratio regimes. This data is commonly generated through computer simulation. For this purpose, our group developed a method to simulate dose reduction for digital breast tomosynthesis. In the previous work, tests were performed in a system that features an amorphous selenium detector with minimal pixel correlation. In the current work, we evaluate the simulation performance in an amorphous silicon system, which yields a relevant pixel correlation. Signal and noise characteristics in real and simulated images were measured using the signal-to-noise ratio (SNR) and the normalized noise power spectrum (NNPS). The simulation method assessment was performed through the average relative error between simulated and real images. The SNR results point to an error of less than 2.5% between the images. The noise correlation influence was verified through the NNPS. The tests pointed to errors up to 55% between the real and simulated images when the correlation kernel is not considered, whereas the error considering the correlation kernel was kept around 5.5%. Therefore, the results show that the correlation kernel is a relevant factor to be considered when simulating amorphous silicon systems.