REMOVABLE SETS FOR LIPSCHITZ HARMONIC FUNCTIONS ON CARNOT GROUPS

Abstract. Let G be a Carnot group with homogeneous dimension Q ≥ 3 and let L be a sub-Laplacian on G. We prove that the critical dimension for removable sets of Lipschitz L-harmonic functions is (Q − 1). Moreover we construct self-similar sets with positive and finite H Q−1 measure which are removable. 1

Traceability of the Hannover FG5X-220 to the SI units

The absolute measurement of g is currently realized through the laser interferometric measurement of a free falling retro-reflector. The Micro-g LaCoste FG5X is a free-fall gravimeter with a laser interferometer in Mach-Zehnder configuration which uses simultaneous time and distance measurements to calculate the absolute value of g. Because the instrument itself contains the necessary working standards for precise time and length measurements, it is considered independent of external references. The timing is kept with a 10MHz rubidium oscillator with a stability of 5x10e-10. The length unit is realized by the laser interferometer. The frequency calibrated and iodine stabilized helium-neon laser has a wavelength of 633 nm and an accuracy of 2.5x10e-11. In 2012 the FG5-220 of the Institut für Erdmessung (IfE) was upgraded to the FG5X-220. The upgrade included a new dropping chamber with a longer free fall and new electronics including a new rubidium oscillator. The metrological traceability to measurement units of the Système International d’unités (SI unit) is ensured by two complementary and successive approaches: the comparison of frequencies with standards of higher order and the comparison of the measured g to a reference measured by absolute gravimeters defined as primary standards within the SI. A number of experiments to test the rubidium oscillator were performed. The oscillator showed a linear drift of 0.2x10e-3 Hz per month (= 0.3 nm/s² per month) in the first 18 months of use. A jump in the frequency of 0.01 Hz (=20 nm/s²) was revealed recently and the drift rate changed to 0.4x10e-3 Hz/month