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Let $\Omega$ and $\Omega'$ be open subsets of a flat $(2,3,5)$-distribution. We show that a $C^1$-smooth contact mapping $f : \Omega \to \Omega'$ is a $C^\infty$-smooth contact mapping. Ultimately, this is a consequence of the rigidity of the associated stratified Lie group (the Tanaka prolongation of the Lie algebra is of finite-type). The conclusion is reached through a careful study of some differential identities satisfied by components of the Pansu-derivative of a $C^1$-smooth contact mapping., Comment: 16 pages

This is the final report submitted to NASA for a Probe-class concept study of the "Farside Array for Radio Science Investigations of the Dark ages and Exoplanets" (FARSIDE), a low radio frequency interferometric array on the farside of the Moon. The design study focused on the instrument, a deployment rover, the lander and base station, and delivered an architecture broadly consistent with the requirements for a Probe mission. This notional architecture consists of 128 dipole antennas deployed across a 10 km area by a rover, and tethered to a base station for central processing, power and data transmission to the Lunar Gateway, or an alternative relay satellite. FARSIDE would provide the capability to image the entire sky each minute in 1400 channels spanning frequencies from 150 kHz to 40 MHz, extending down two orders of magnitude below bands accessible to ground-based radio astronomy. The lunar farside can simultaneously provide isolation from terrestrial radio frequency interference, auroral kilometric radiation, and plasma noise from the solar wind. This would enable near-continuous monitoring of the nearest stellar systems in the search for the radio signatures of coronal mass ejections and energetic particle events, and would also detect the magnetospheres for the nearest candidate habitable exoplanets. Simultaneously, FARSIDE would be used to characterize similar activity in our own solar system, from the Sun to the outer planets. Through precision calibration via an orbiting beacon, and exquisite foreground characterization, FARSIDE would also measure the Dark Ages global 21-cm signal at redshifts from 50-100. It will also be a pathfinder for a larger 21-cm power spectrum instrument by carefully measuring the foreground with high dynamic range., Comment: 50 pages, NASA Probe final study report. arXiv admin note: text overlap with arXiv:1907.05407

Let $\mathbb{H}$ be the sub-Riemannian Heisenberg group. That $\mathbb{H}$ supports a rich family of quasiconformal mappings was demonstrated by Kor\'{a}nyi and Reimann using the so-called flow method. Here we supply further evidence of the flexible nature of this family, constructing quasiconformal mappings with extreme behavior on small sets. More precisely, we establish criteria to determine when a given logarithmic potential $\Lambda$ on $\mathbb{H}$ is such that there exists a quasiconformal mapping of $\mathbb{H}$ with Jacobian comparable to $e^{2\Lambda}$ (so that the Jaobian is zero or infinity at the same points as $e^{2\Lambda}$). When $\Lambda$ is continuous and meets the criteria, we show the canonical (sub-Riemannian) metric $g_0$ and the weighted metric $g = e^\Lambda g_0$ generate bi-Lipschitz equivalent distance functions. These results rest on an extension to the theory of quasiconformal flows on $\mathbb{H}$ and constructions that adapt the iterative method of Bonk, Heinonen, and Saksman., Comment: 71 pages

We give a new proof for the $C^\infty$ regularity of $C^2$ smooth conformal mappings of the sub-Riemannian Heisenberg group. Our proof avoids any use of nonlinear potential theory and relies only on hypoellipticity of H\"ormander operators and quasiconformal flows. This approach is inspired by prior work of Sarvas and Liu., Comment: 10 pages

NASA is responding to the growing interest in, andcapabilities of, small satellites for science applications with an increasingnumber and frequency of Announcements of Opportunityfor small satellite space missions. Estimating the probabilitythat these mission concepts will fit within the small cost capsof these opportunities is largely driven by the probability thatone of the burgeoning number of small satellite providers will beable to meet the payload’s accommodation requirements withinthe budget for the spacecraft. JPL has collected a databasecontaining technical specifications and cost of commerciallyavailable Smallsat buses across various vendors. The primarypurpose of the database is for use in JPL’s Team X architecturestudies to inform cost estimates of a spacecraft bus which fitsthe customer’s technical requirements for their payload andmission. Customer needs are often unique and don’t alignperfectly with an off-the-shelf commercial spacecraft bus, whichmotivates the need to develop a cost model across the continuoustechnical parameter space.Al’s Bus Cost Distribution Estimator (ABCDE) uses Gaussianprocess regression (GPR) to predict commercial Smallsat spacecraftbus cost based on a subset of a customer’s technicalrequirements (payload mass, payload power, delta V, pointingcontrol, and downlink rate). GPR is implemented in ABCDE asa Bayesian method which fits an implied multivariate regressionon the technical parameters and uses kriging to intentionally“overfit” the residuals. Overfitting the residuals allows costestimates to collapse in uncertainty closer to the data pointswhile maintaining larger uncertainty intervals in regions of parameterspace with fewer data records. The data used to fit thismodel is sensitive and represents cost estimates for off-the-shelfcommercial buses. GPR simultaneously protects the sensitivityof the database and uses the sparse nature of the database toaccount for uncertainty in cost in a useful way. For a givenset of customer technical requirements, the tool provides a costestimate distribution, the percentiles of which can be interpretedas a confidence level of finding a commercial bus under a specifiedcost cap. ABCDE dramatically pushes the boundaries ofspacecraft cost estimation models due to its Bayesian methodology(accounting for the maximum uncertainty in the underlyingregression), the mathematically advanced kriging methodology,and the novelty of its application in Team X architecture tradestudies.

The use of aerocapture to provide the delta-V needed to capture a spacecraft into orbit can provide significant fuel savings compared to a propulsive orbit insertion. The aerocapture guidance method described in this paper uses drag modulation, which deploys a drag skirt in the atmosphere until the amount of delta-V needed to capture into the desired orbit is obtained, at which time the drag skirt is separated from the spacecraft. By modulating the time of drag skirt jettison, the vehicle can target a specific orbit apoapsis in the presence of uncertainties such as entry targeting errors and unknown atmospheric conditions.

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The paper presents an overview of the ground rules, assumptions, methodology, operations model, element designs, base site plan, and quantitative findings. These findings include the performance of various regolith and ice resource utilization schemes as a function of base location and lunar surface parameters. The paper closes with short lists of the highest priority experiments and demonstrations needed on the lunar surface to retire key planning unknowns.