The Newtonian gravitational constant, G, is one of the most fundamental constants of nature, but we still do not have an accurate value for it. Despite two centuries of experimental effort, the value of G remains the least precisely known of the fundamental constants. A discrepancy of up to 0.05 per cent in recent determinations of G suggests that there may be undiscovered systematic errors in the various existing methods. One way to resolve this issue is to measure G using a number of methods that are unlikely to involve the same systematic effects. Here we report two independent determinations of G using torsion pendulum experiments with the time-of-swing method and the angular-acceleration-feedback method. We obtain G values of 6.674184 × 10 and 6.674484 × 10 cubic metres per kilogram per second squared, with relative standard uncertainties of 11.64 and 11.61 parts per million, respectively. These values have the smallest uncertainties reported until now, and both agree with the latest recommended value within two standard deviations.

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Explosions near the Earth's surface excite both seismic ground motions and atmospheric overpressure. The energy transferred to the ground and atmosphere from a near-surface explosion depends on yield (W) as well as the height-of-burst/ depth-of-burial (HOB/DOB) for above/belowground emplacements. We report analyses of seismic and overpressure motions from the Humble Redwood series of low-yield, near-surface chemical explosions with the aim of developing quantitative models of energy partitioning and a methodology to estimate W and HOB/DOB. The effects of yield, HOB, and range on amplitudes can be cast into separable functions of range and HOB scaled by yield. We find that displacement of the initial P wave and the integral of the positive overpressure (impulse) are diagnostic of W and HOB with minimal scatter. An empirical model describing the dependence of seismic and air-blast measurements on W, HOB/DOB, and range is determined and model parameters are found by regression. We find seismic amplitudes for explosions of a given yield emplaced at or above the surface are reduced by a factor of 3 relative to fully contained explosions below ground. Air-blast overpressure is reduced more dramatically, with impulse reduced by a factor of 100 for deeply buried explosions relative to surface blasts. Our signal models are used to invert seismic and overpressure measurements for W and HOB and we find good agreement (W errors < 30%, HOB within meters) with groundtruth values for four noncircular validation tests. Although there is a trade-off between W and HOB for a single seismic or overpressure measurement, the use of both measurement types allows us to largely break this trade-off and better constrain W and HOB. However, both models lack resolution of HOB for aboveground explosions.

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HI and D 2 are coexpanded into a vacuum chamber. The photolysis of HI at 212.8 nm initiates the HϩD 2 reaction. The HD͑vϭ4, JЈϭ3͒ velocity distribution is determined by analyzing the time-of-flight profile of HD ϩ ions produced by delayed pulsed field ionization of long-lived Rydberg states. The angular distribution is deduced using the law of cosines ͑photoloc technique͒.

SignificanceThe Bôcher theorem for fractional Laplacian extends the classical Bôcher theorem with a unified proof that can be adapted in other situations. Our distributional approach reduces the regularity requirement and connects the Bôcher theorem directly with the corresponding maximum principles. These maximum principles derived are simple and basic with many potential applications.

We describe an apparatus designed to measure the velocity distribution of products of photoinitiated bulb reactions. A photolytic precursor AX and reactant BC are coexpanded into a vacuum chamber. A photolysis laser initiates the reaction sequence AX + hv -A + X, A + BC -AB + C. The C product is detected by sub-Doppler (1 + 1 + 1) resonance-enhanced multiphoton ionization (2D-REMPI) to yield its three-dimensional velocity distribution. If this technique were to be applied to the AB(v',J') product, it would be possible to measure the alignment dependenceof the state-to-state differential cross section. We present the experimentally determined velocity distribution of H atoms from the photolysis of H I and D atoms from the reaction H + D2 -H D + D, and we show that our measurements are consistent with previous studies.

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