Weighted Birkhoff Averages and the Parameterization Method

This work provides a systematic recipe for computing accurate high order Fourier expansions of quasiperiodic invariant circles in area preserving maps. The recipe requires only a finite data set sampled from the quasiperiodic circle. Our approach, being based on the parameterization method, uses a Newton scheme to iteratively solve a conjugacy equation describing the invariant circle. A critical step in properly formulating the conjugacy equation is to determine the rotation number of the quasiperiodic subsystem. For this we exploit a the weighted Birkhoff averaging method. This approach facilities accurate computation of the rotation number given nothing but the already mentioned orbit data. The weighted Birkhoff averages also facilitate the computation of other integral observables like Fourier coefficients of the parameterization of the invariant circle. Since the parameterization method is based on a Newton scheme, we only need to approximate a small number of Fourier coefficients with low accuracy to find a good enough initial approximation so that Newton converges. Moreover, the Fourier coefficients may be computed independently, so we can sample the higher modes to guess the decay rate of the Fourier coefficients. This allows us to choose, a-priori, an appropriate number of modes in the truncation. We illustrate the utility of the approach for explicit example systems including the area preserving Henon map and the standard map. We present example computations for invariant circles with period as low as 1 and up to more than 100. We also employ a numerical continuation scheme to compute large numbers of quasiperiodic circles in these systems. During the continuation we monitor the Sobolev norm of the Parameterization to automatically detect the breakdown of the family.Comment: 38 pages, 15 figure

Design of a Mechanical NaK Pump for Fission Space Power Systems

Alkali liquid metal cooled fission reactor concepts are under development for mid-range spaceflight power requirements. One such concept utilizes a sodium-potassium eutectic (NaK) as the primary loop working fluid. Traditionally, linear induction pumps have been used to provide the required flow and head conditions for liquid metal systems but can be limited in performance. This paper details the design, build, and check-out test of a mechanical NaK pump. The pump was designed to meet reactor cooling requirements using commercially available components modified for high temperature NaK service

Cheon\u27s algorithm, pairing inversion and the discrete logarithm problem

We relate the fixed argument pairing inversion problems (FAPI) and the discrete logarithm problem on an elliptic curve. This is done using the reduction from the DLP to the Diffie-Hellman problem developed by Boneh, Lipton, Maurer and Wolf. This approach fails when only one of the FAPI problems can be solved. In this case we use Cheon\u27s algorithm to get a reduction

Moir\'e band structures of twisted phosphorene bilayers

We report on the theoretical electronic spectra of twisted phosphorene bilayers exhibiting moir\'e patterns, as computed by means of a continuous approximation to the moir\'e superlattice Hamiltonian. Our model is constructed by interpolating between effective $\Gamma$-point conduction- and valence-band Hamiltonians for the different stacking configurations approximately realized across the moir\'e supercell, formulated on symmetry grounds. We predict the realization of three distinct regimes for $\Gamma$-point electrons and holes at different twist angle ranges: a Hubbard regime for small twist angles $\theta < 2^\circ$, where the electronic states form arrays of quantum-dot-like states, one per moir\'e supercell. A Tomonaga-Luttinger regime at intermediate twist angles $2^\circ < \theta \lesssim 10^\circ$, characterized by the appearance of arrays of quasi-1D states. Finally, a ballistic regime at large twist angles $\theta \gtrsim 10^\circ$, where the band-edge states are delocalized, with dispersion anisotropies modulated by the twist angle. Our method correctly reproduces recent results based on large-scale ab initio calculations at a much lower computational cost, and with fewer restrictions on the twist angles considered.Comment: 20 pages, including 10 figures and 5 appendice

Pairings on hyperelliptic curves with a real model

We analyse the efficiency of pairing computations on hyperelliptic curves given by a real model using a balanced divisor at infinity. Several optimisations are proposed and analysed. Genus two curves given by a real model arise when considering pairing friendly groups of order dividing $p^{2}-p+1$. We compare the performance of pairings on such groups in both elliptic and hyperelliptic versions. We conclude that pairings can be efficiently computable in real models of hyperelliptic curves

Securing the legacy of TESS through the care and maintenance of TESS planet ephemerides

Much of the science from the exoplanets detected by the TESS mission relies on precisely predicted transit times that are needed for many follow-up characterization studies. We investigate ephemeris deterioration for simulated TESS planets and find that the ephemerides of 81% of those will have expired (i.e. 1$\sigma$ mid-transit time uncertainties greater than 30 minutes) one year after their TESS observations. We verify these results using a sample of TESS planet candidates as well. In particular, of the simulated planets that would be recommended as JWST targets by Kempton et al. (2018), $\sim$80% will have mid-transit time uncertainties $>$ 30 minutes by the earliest time JWST would observe them. This rapid deterioration is driven primarily by the relatively short time baseline of TESS observations. We describe strategies for maintaining TESS ephemerides fresh through follow-up transit observations. We find that the longer the baseline between the TESS and the follow-up observations, the longer the ephemerides stay fresh, and that 51% of simulated primary mission TESS planets will require space-based observations. The recently-approved extension to the TESS mission will rescue the ephemerides of most (though not all) primary mission planets, but the benefits of these new observations can only be reaped two years after the primary mission observations. Moreover, the ephemerides of most primary mission TESS planets (as well as those newly discovered during the extended mission) will again have expired by the time future facilities such as the ELTs, Ariel and the possible LUVOIR/OST missions come online, unless maintenance follow-up observations are obtained.Comment: 16 pages, 10 figures, accepted to AJ; main changes are cross-checking results against the sample of real TOIs, and addressing the impact of the TESS extended missio