WORK-LIFE: AN INTERPLAY OF ISSUES

Labor and Human Capital,

Noncommutativity in the analysis of piecewise discrete-time dynamical systems

In this paper, we present a new method for the analysis of piecewise dynamical systems that are similar to the Collatz conjecture in regard to certain properties of the commutator of their sub-functions. We use the fact that the commutator of polynomials $E(n)=n/2$ and $O(n)=(3n+1)/2$ is constant to study rearrangements of compositions of $E(n)$ and $O(n)$. Our main result is that for any positive rational number $n$, if $(E^{e_1} \circ O^{o_1} \circ E^{e_2} \circ \dotsb \circ O^{o_l} \circ E^{e_{l+1}})(n)=1$, then $(E^{e_1} \circ O^{o_1} \circ E^{e_2} \circ \dotsb \circ O^{o_l} \circ E^{e_{l+1}})(n) = \lceil(E^{e_1 + \dotsb + e_{l+1}} \circ O^{o_1 + \dotsb + o_{l}})(n)\rceil$, where exponentiation is used to denote repeated composition and $e_i$ and $o_i$ are positive integers. Composition sequences of this form have significance in the context of the Collatz conjecture. The techniques used to derive this result can be used to produce similar results for a wide variety of repeatedly composed piecewise functions.Comment: 7 page

The total mass of the Large Magellanic Cloud from its perturbation on the Orphan stream

In a companion paper by Koposov et al., RR Lyrae from \textit{Gaia} Data Release 2 are used to demonstrate that stars in the Orphan stream have velocity vectors significantly misaligned with the stream track, suggesting that it has received a large gravitational perturbation from a satellite of the Milky Way. We argue that such a mismatch cannot arise due to any realistic static Milky Way potential and then explore the perturbative effects of the Large Magellanic Cloud (LMC). We find that the LMC can produce precisely the observed motion-track mismatch and we therefore use the Orphan stream to measure the mass of the Cloud. We simultaneously fit the Milky Way and LMC potentials and infer that a total LMC mass of $1.38^{+0.27}_{-0.24} \times10^{11}\,\rm{M_\odot}$ is required to bend the Orphan Stream, showing for the first time that the LMC has a large and measurable effect on structures orbiting the Milky Way. This has far-reaching consequences for any technique which assumes that tracers are orbiting a static Milky Way. Furthermore, we measure the Milky Way mass within 50 kpc to be $3.80^{+0.14}_{-0.11}\times10^{11} M_\odot$. Finally, we use these results to predict that, due to the reflex motion of the Milky Way in response to the LMC, the outskirts of the Milky Way's stellar halo should exhibit a bulk, upwards motion.Comment: 17 pages, 11 figures. Updated to version accepted to MNRAS after minor revisio