Large Steklov eigenvalues via homogenisation on manifolds

Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace eigenvalues of that manifold. The domains are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero. We use this relationship to construct manifolds that have large Steklov eigenvalues. In dimension two, and with constant weight equal to 1, we prove that Kokarev’s upper bound of 8\pi for the first nonzero normalised Steklov eigenvalue on orientable surfaces of genus 0 is saturated. For other topological types and eigenvalue indices, we also obtain lower bounds on the best upper bound for the eigenvalue in terms of Laplace maximisers. For the first two eigenvalues, these lower bounds become equalities. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov eigenfunctions and with area strictly larger than 2\pi. This was previously thought to be impossible. We provide numerical evidence that some of the already known examples of free boundary minimal surfaces have these properties and also exhibit simulations of new free boundary minimal surfaces of genus zero in the unit ball with even larger area. We prove that the first nonzero Steklov eigenvalue of all these examples is equal to 1, as a consequence of their symmetries and topology, so that they are consistent with a general conjecture by Fraser and Li. In dimension three and larger, we prove that the isoperimetric inequality of Colbois–El Soufi–Girouard is sharp and implies an upper bound for weighted Laplace eigenvalues. We also show that in any manifold with a fixed metric, one can construct by varying the weight a domain with connected boundary whose first nonzero normalised Steklov eigenvalue is arbitrarily large

Recertification of the air and methane storage vessels at the Langley 8-foot high-temperature structures tunnel

This center operates a number of sophisticated wind tunnels in order to fulfill the needs of its researchers. Compressed air, which is kept in steel storage vessels, is used to power many of these tunnels. Some of these vessels have been in use for many years, and Langley is currently recertifying these vessels to insure their continued structural integrity. One of the first facilities to be recertified under this program was the Langley 8-foot high-temperature structures tunnel. This recertification involved (1) modification, hydrotesting, and inspection of the vessels; (2) repair of all relevant defects; (3) comparison of the original design of the vessel with the current design criteria of Section 8, Division 2, of the 1974 ASME Boiler and Pressure Vessel Code; (4) fracture-mechanics, thermal, and wind-induced vibration analyses of the vessels; and (5) development of operating envelopes and a future inspection plan for the vessels. Following these modifications, analyses, and tests, the vessels were recertified for operation at full design pressure (41.4 MPa (6000 psi)) within the operating envelope developed

Pointwise Bounds for Steklov Eigenfunctions

Let (Ω,g) be a compact, real-analytic Riemannian manifold with real-analytic boundary ∂Ω. The harmonic extensions of the boundary Dirichlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfunctions decay exponentially into the interior in terms of the Dirichlet-to-Neumann eigenvalues and give a sharp rate of decay to first order at the boundary. The proof uses the Poisson representation for the Steklov eigenfunctions combined with sharp h-microlocal concentration estimates for the boundary Dirichlet-to-Neumann eigenfunctions near the cosphere bundle S∗∂Ω. These estimates follow from sharp estimates on the concentration of the FBI transforms of solutions to analytic pseudodifferential equations Pu=0 near the characteristic set {σ(P)=0}

Kepler Presearch Data Conditioning I - Architecture and Algorithms for Error Correction in Kepler Light Curves

Kepler provides light curves of 156,000 stars with unprecedented precision. However, the raw data as they come from the spacecraft contain significant systematic and stochastic errors. These errors, which include discontinuities, systematic trends, and outliers, obscure the astrophysical signals in the light curves. To correct these errors is the task of the Presearch Data Conditioning (PDC) module of the Kepler data analysis pipeline. The original version of PDC in Kepler did not meet the extremely high performance requirements for the detection of miniscule planet transits or highly accurate analysis of stellar activity and rotation. One particular deficiency was that astrophysical features were often removed as a side-effect to removal of errors. In this paper we introduce the completely new and significantly improved version of PDC which was implemented in Kepler SOC 8.0. This new PDC version, which utilizes a Bayesian approach for removal of systematics, reliably corrects errors in the light curves while at the same time preserving planet transits and other astrophysically interesting signals. We describe the architecture and the algorithms of this new PDC module, show typical errors encountered in Kepler data, and illustrate the corrections using real light curve examples.Comment: Submitted to PASP. Also see companion paper "Kepler Presearch Data Conditioning II - A Bayesian Approach to Systematic Error Correction" by Jeff C. Smith et a

The Steklov spectrum of cuboids

The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension d ≥ 3. Apart from the standard Weyl term, we calculate explicitly the second term in the asymptotics, capturing the contribution of the (d - 2) - dimensional facets of a cuboid. Our approach is based on lattice counting techniques. While this strategy is similar to the one used for the Dirichlet Laplacian, the Steklov case carries additional complications. In particular, it is not clear how to establish directly the completeness of the system of Steklov eigenfunctions admitting separation of variables. We prove this result using a family of auxiliary Robin boundary value problems. Moreover, the correspondence between the Steklov eigenvalues and lattice points is not exact, and hence more delicate analysis is required to obtain spectral asymptotics. Some other related results are presented, such as an isoperimetric inequality for the first Steklov eigenvalue, a concentration property of high frequency Steklov eigenfunctions and applications to spectral determination of cuboids

Kepler Presearch Data Conditioning II - A Bayesian Approach to Systematic Error Correction

With the unprecedented photometric precision of the Kepler Spacecraft, significant systematic and stochastic errors on transit signal levels are observable in the Kepler photometric data. These errors, which include discontinuities, outliers, systematic trends and other instrumental signatures, obscure astrophysical signals. The Presearch Data Conditioning (PDC) module of the Kepler data analysis pipeline tries to remove these errors while preserving planet transits and other astrophysically interesting signals. The completely new noise and stellar variability regime observed in Kepler data poses a significant problem to standard cotrending methods such as SYSREM and TFA. Variable stars are often of particular astrophysical interest so the preservation of their signals is of significant importance to the astrophysical community. We present a Bayesian Maximum A Posteriori (MAP) approach where a subset of highly correlated and quiet stars is used to generate a cotrending basis vector set which is in turn used to establish a range of "reasonable" robust fit parameters. These robust fit parameters are then used to generate a Bayesian Prior and a Bayesian Posterior Probability Distribution Function (PDF) which when maximized finds the best fit that simultaneously removes systematic effects while reducing the signal distortion and noise injection which commonly afflicts simple least-squares (LS) fitting. A numerical and empirical approach is taken where the Bayesian Prior PDFs are generated from fits to the light curve distributions themselves.Comment: 43 pages, 21 figures, Submitted for publication in PASP. Also see companion paper "Kepler Presearch Data Conditioning I - Architecture and Algorithms for Error Correction in Kepler Light Curves" by Martin C. Stumpe, et a

Software Assists in Responding to Anomalous Conditions

Fault Induced Document Retrieval Officer (FIDO) is a computer program that reduces the need for a large and costly team of engineers and/or technicians to monitor the state of a spacecraft and associated ground systems and respond to anomalies. FIDO includes artificial-intelligence components that imitate the reasoning of human experts with reference to a knowledge base of rules that represent failure modes and to a database of engineering documentation. These components act together to give an unskilled operator instantaneous expert assistance and access to information that can enable resolution of most anomalies, without the need for highly paid experts. FIDO provides a system state summary (a configurable engineering summary) and documentation for diagnosis of a potentially failing component that might have caused a given error message or anomaly. FIDO also enables high-level browsing of documentation by use of an interface indexed to the particular error message. The collection of available documents includes information on operations and associated procedures, engineering problem reports, documentation of components, and engineering drawings. FIDO also affords a capability for combining information on the state of ground systems with detailed, hierarchically-organized, hypertext- enabled documentation

Verification of the Kepler Input Catalog from Asteroseismology of Solar-type Stars

We calculate precise stellar radii and surface gravities from the asteroseismic analysis of over 500 solar-type pulsating stars observed by the Kepler space telescope. These physical stellar properties are compared with those given in the Kepler Input Catalog (KIC), determined from ground-based multi-color photometry. For the stars in our sample, we find general agreement but we detect an average overestimation bias of 0.23 dex in the KIC determination of log (g) for stars with log (g)_KIC > 4.0 dex, and a resultant underestimation bias of up to 50% in the KIC radii estimates for stars with R_KIC < 2 R sun. Part of the difference may arise from selection bias in the asteroseismic sample; nevertheless, this result implies there may be fewer stars characterized in the KIC with R ~ 1 R sun than is suggested by the physical properties in the KIC. Furthermore, if the radius estimates are taken from the KIC for these affected stars and then used to calculate the size of transiting planets, a similar underestimation bias may be applied to the planetary radii.Comment: Published in The Astrophysical Journal Letter