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The Brillouin sphere is defined as the smallest sphere, centered at the origin of the geocentric coordinate system, that incorporates all the condensed matter composing the planet. The Brillouin sphere touches the Earth at a single point, and the radial line that begins at the origin and passes through that point is called the singular radial line. For about 60 years there has been a persistent anxiety about whether or not a spherical harmonic (SH) expansion of the external gravitational potential,V, will converge beneath the Brillouin sphere. Recently, it was proven that the probability of such convergence is zero. One of these proofs provided an asymptotic relation, called Costin’s formula, for the upper bound,E_N, on the absolute value of the prediction error,e_N, of a SH series model, $V_N(\theta ,\lambda ,r)$, truncated at some maximum degree, $N=n_{max}$. When the SH series is restricted to (or projected onto) a particular radial line, it reduces to a Taylor series (TS) in $1/r$. Costin’s formula is $E_N\simeq B{N}^{-b}(R/r{)}^N$, whereRis the radius of the Brillouin sphere. This formula depends on two positive parameters:b, which controls the decay of error amplitude as a function ofNwhenris fixed, and a scale factorB. We show here that Costin’s formula derives from a similar asymptotic relation for the upper bound,A_non the absolute value of the TS coefficients,a_n, for the same radial line. This formula, $A_n\simeq K{n}^{-k}$, depends on degree,n, and two positive parameters,kandK, that are analogous tobandB. We use synthetic planets, for which we can compute the potential,V, and also the radial component of gravitational acceleration, $g_r=\partial V/\partial r$, to hundreds of significant digits, to validate both of these asymptotic formulas. Let superscriptVrefer to asymptotic parameters associated with the coefficients and prediction errors for gravitational potential, and superscriptgto the coefficients and predictions errors associated withg_r. For polyhedral planets of uniform density we show that $b^V=k^V=7/2$and $b^g=k^g=5/2$almost everywhere. We show that the frequency of oscillation (around zero) of the TS coefficients and the series prediction errors, for a given radial line, is controlled by the geocentric angle,α, between that radial line and the singular radial line. We also derive useful identities connecting $K^V,B^V,K^g$, andB^g. These identities are expressed in terms of quotients of the various scale factors. The only other quantities involved in these identities areαandR. The phenomenology of ‘series divergence’ and prediction error (whenr < R) can be described as a function of the truncation degree,N, or the depth,d, beneath the Brillouin sphere. For a fixed $r\text{⩽}R$, asNincreases from very low values, the upper error boundE_Nshrinks until it reaches its minimum (best) value whenNreaches some particular or optimum value, $N_{\mathrm{opt}}$. When $N>N_{\mathrm{opt}}$, prediction error grows asNcontinues to increase. Eventually, when $N\gg N_{\mathrm{opt}}$, prediction errors increase exponentially with risingN. If we fix the value ofNand allow $R/r$to vary, then we find that prediction error in free space beneath the Brillouin sphere increases exponentially with depth,d, beneath the Brillouin sphere. Because $b^g=b^V-1$everywhere, divergence driven prediction error intensifies more rapidly forg_rthan forV, both in terms of its dependence onNandd. If we fix bothNandd, and focus on the ‘lateral’ variations in prediction error, we observe that divergence and prediction error tend to increase (as doesB) as we approach high-amplitude topography. 

In recent years, there has been a growing interest in using tandem foils to mimic and study fish swimming, and to inform underwater vehicle design. Though much effort has been put to understanding the propulsion mechanisms of a tandem-foil system, the stability of such a system and the mechanisms for maintaining it remain an open question. In this study, a 3-foil system in an in-line configuration is used towards understanding the hydrodynamics of lateral stability. The foils actively pitch with varying phase. To quantify lateral force oscillation, the standard deviation of the lateral force, 𝝈𝝈𝒀𝒀, calculated over one typical flapping cycle is used, to account for the amount of variation in the lateral force experienced by the system of 3 foils. The higher the standard deviation, the more the spread in the lateral force cycle data, the more lateral momentum exchanged between the flow and the foils, and the less stable the system is. Through phase variations, it is found that the lateral force is minimized when the phases of the three foils are approximately, though not exactly, evenly distributed. The least stable system is found to be the one with the foils all in phase. Systems that are more laterally stable are found to tend to have narrower envelopes of regions around the foils with high momentum. Near-wake of the foils, the envelopes of stable systems are also found to have pronounced convergent sections, whereas the envelope of the less stable systems are found to diverge without much interruption. In the far wake, coherent, singular thrust jets, along with orderly 2-S vortices are found to form in the two best performing cases. In less stable cases, the thrust jets are found to be branched. Corresponding to the width of the high-momentum envelopes, lateral jets are found to exist in the gaps between neighboring foils, the strengths of which vary based on stability, with the lateral jets being more pronounced in the less stable cases (cases with high amount of lateral force oscillation). Peak lateral forces are found to coincide with moments of pressure gradient build-up across the foils. The pressure-driven flow near the trailing edge of the foils then creates trailing-edge vortices, and correspondingly, lateral gap flows. Moments of peak and plateau lateral force on an individual foil in the system are found to coincide with the initiation and shedding of trailing-edge vortices, respectively. The formation of trailing-edge vortices, lateral jets and cross-stream flows in gaps are closely intertwined, and all are 1. Indicative of large lateral momentum oscillation, and 2. The results of pressure gradient build-up across foils. 

Recent technology development of logic devices based on 2-D semiconductors such as MoS2, WS2, and WSe2 has triggered great excitement, paving the way to practical applications. Making low-resistance p-type contacts to 2-D semiconductors remains a critical challenge. The key to addressing this challenge is to find high-work function metallic materials which also introduce minimal metal-induced gap states (MIGSs) at the metal/semiconductor interface. In this work, we perform a systematic computational screening of novel metallic materials and their heterojunctions with monolayer WSe2 based on ab initio density functional theory and quantum device simulations. Two contact strategies, van der Waals (vdW) metallic contact and bulk semimetallic contact, are identified as promising solutions to achieving Schottky-barrier-free and low-contact-resistance p-type contacts for WSe2 p-type field-effect transistor (pFETs). Good candidates of p-type contact materials are found based on our screening criteria, including 1H-NbS2, 1H-TaS2, and 1T-TiS2 in the vdW metal category, as well as Co3Sn2S2 and TaP in the bulk semimetal category. Simulations of these new p-type contact materials suggest reduced MIGS, less Fermi-level pinning effect, negligible Schottky barrier height and small contact resistance (down to 20 Ωμm )