Rapid and Accurate Computation of Invariant Tori, Manifolds, and Connections Near Mean Motion Resonances in the Periodically Perturbed PCRTBP
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In March and April 2021, buoys were deployed in the Beaufort Sea, Arctic Ocean, to measure sea-ice horizontal deformation over spatial scales that had not been previously achieved. Geodetic-quality position measurements allowed measurements of strain-rate over lengths from about 200 m to 2 km. Conventional ice-drifters extended spatial coverage up to about 100 km. Past studies find there is multi-fractal behavior for horizontal sea-ice deformation from 10 to 1000 km. Our results demonstrate that such behavior does not hold when including spatial scales below 10 km. We find that sea-ice deformation is not scale in variant between the scale of individual sea-ice floes and aggregates of floes. Therefore, we cannot expect the same physical laws or forcing to describe sea-ice kinematics over these regimes, nor can we assume log-log linear behavior for mean deformation. Using this scaling behavior as a metric to validate models that resolve sea ice floes and their interactions is hence not recommended.
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The Muon g-2 experiment at Fermilab (E989) aims to measure the anomalous magnetic moment of the muon, $a_{\mu}= (g-2)/2$, to a groundbreaking precision of $140$ ppb, obtaining a near four-fold increase in precision over the previous experiment, E821, at the Brookhaven National Laboratory (BNL). The value of $a_{\mu}$ from BNL currently differs from the Standard Model prediction by $\sim 3.7$ standard deviations, suggesting the potential for new physics and therefore, motivating a new experiment.Because the theory predicts this number with high precision, testing the g-factor through experiment provides a stringent test of the SM and can suggest physics beyond the Standard Model. The goal of the Fermilab Muon $g-2$ experiment is to increase the statistical precision by more than a factor of 20 and reduce systematic errors by a factor of 3. By measuring muon precession rate ($\omega_a$) in an external magnetic field, the anomalous magnetic moment will be calculated. This is an incredibly challenging experiment with a unique opportunity to provide new insight into nature. \\ The $g-2$ data also provides a great opportunity for setting the most stringent limits on some of the Standard Model Extension CPT Lorentz violating (LV) parameters in the muon sector. One of the CPT and Lorentz violating signatures that we can look for using $g-2$ data is a sidereal variation of $\omega_a(t)$. Extensive simulation studies confirm that the sensitivity regarding the sidereal varation roughly scales with $\omega_a$ uncertainty. Hence, the $g-2$ experiment at FNAL should be able to reach limits of $\sim 5\times10^{-25}$ GeV. Because the CPT and LV analyses are essentially studies of variations in $\omega_a$ as a function of time and charge, performing an $\omega_a$ analysis sets the stage for the CPT and LV measurement. This dissertation focuses on the methodology of a fully functioning framework and analyzing the Fermilab Muon $g - 2$ Run 2 data containing $\sim 11$ billion events above an energy threshold of $1.7$~GeV.
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