Cobb-Douglas Utility - Eventually!

Consider the following two opinions, both of which can be found in the literature of consumer demand systems: (a) As the real income of a consumer becomes indefinitely large, re-mixing the consumption bundle becomes irrelevant: having chosen the ultimately satisfying budget shares at any given set of relative prices, the superlatively wealthy continue to allocate additional income in the same proportions. With very large and increasing per capita income, ultimately the utility function becomes indistinguishable from Cobb-Douglas. (b) Consumer demand systems in which the income elasticities monotonically approach one (from above, in the case of luxuries; from below, in the case of necessities) are unsatisfactory both theoretically and empirically. For instance, a necessity with a low (consumer demand system; applied general equilibrium; separability; implicitly directly additive preferences; effectively global regularity; Cobb-Douglas, calibration; AIDADS.

About ergodicity in the family of limacon billiards

By continuation from the hyperbolic limit of the cardioid billiard we show that there is an abundance of bifurcations in the family of limacon billiards. The statistics of these bifurcation shows that the size of the stable intervals decreases with approximately the same rate as their number increases with the period. In particular, we give numerical evidence that arbitrarily close to the cardioid there are elliptic islands due to orbits created in saddle node bifurcations. This shows explicitly that if in this one parameter family of maps ergodicity occurs for more than one parameter the set of these parameter values has a complicated structure.Comment: 17 pages, 9 figure

A spectral survey of an ultra-hot Jupiter: Detection of metals in the transmission spectrum of KELT-9 b

Context: KELT-9 b exemplifies a newly emerging class of short-period gaseous exoplanets that tend to orbit hot, early type stars - termed ultra-hot Jupiters. The severe stellar irradiation heats their atmospheres to temperatures of $\sim 4,000$ K, similar to the photospheres of dwarf stars. Due to the absence of aerosols and complex molecular chemistry at such temperatures, these planets offer the potential of detailed chemical characterisation through transit and day-side spectroscopy. Studies of their chemical inventories may provide crucial constraints on their formation process and evolution history. Aims: To search the optical transmission spectrum of KELT-9 b for absorption lines by metals using the cross-correlation technique. Methods: We analyse 2 transits observed with the HARPS-N spectrograph. We use an isothermal equilibrium chemistry model to predict the transmission spectrum for each of the neutral and singly-ionized atoms with atomic numbers between 3 and 78. Of these, we identify the elements that are expected to have spectral lines in the visible wavelength range and use those as cross-correlation templates. Results: We detect absorption of Na I, Cr II, Sc II and Y II, and confirm previous detections of Mg I, Fe I, Fe II and Ti II. In addition, we find evidence of Ca I, Cr I, Co I, and Sr II that will require further observations to verify. The detected absorption lines are significantly deeper than model predictions, suggesting that material is transported to higher altitudes where the density is enhanced compared to a hydrostatic profile. There appears to be no significant blue-shift of the absorption spectrum due to a net day-to-night side wind. In particular, the strong Fe II feature is shifted by $0.18 \pm 0.27$ km~s$^{-1}$, consistent with zero. Using the orbital velocity of the planet we revise the steller and planetary masses and radii.Comment: Submitted to Astronomy and Astrophysics on January 18, 2019. Accepted on May 3, 2019. 26 pages, 11 figure

Designing theoretically-informed implementation interventions

Clinical and health services research is continually producing new findings that may contribute to effective and efficient patient care. However, the transfer of research findings into practice is unpredictable and can be a slow and haphazard process. Ideally, the choice of implementation strategies would be based upon evidence from randomised controlled trials or systematic reviews of a given implementation strategy. Unfortunately, reviews of implementation strategies consistently report effectiveness some, but not all of the time; possible causes of this variation are seldom reported or measured by the investigators in the original studies. Thus, any attempts to extrapolate from study settings to the real world are hampered by a lack of understanding of the effects of key elements of individuals, interventions, and the settings in which they were trialled. The explicit use of theory offers a way of addressing these issues and has a number of advantages, such as providing: a generalisable framework within which to represent the dimensions that implementation studies address, a process by which to inform the development and delivery of interventions, a guide when evaluating, and a way to allow for an exploration of potential causal mechanisms. However, the use of theory in designing implementation interventions is methodologically challenging for a number of reasons, including choosing between theories and faithfully translating theoretical constructs into interventions. The explicit use of theory offers potential advantages in terms of facilitating a better understanding of the generalisability and replicability of implementation interventions. However, this is a relatively unexplored methodological area

Linear stability in billiards with potential

A general formula for the linearized Poincar\'e map of a billiard with a potential is derived. The stability of periodic orbits is given by the trace of a product of matrices describing the piecewise free motion between reflections and the contributions from the reflections alone. For the case without potential this gives well known formulas. Four billiards with potentials for which the free motion is integrable are treated as examples: The linear gravitational potential, the constant magnetic field, the harmonic potential, and a billiard in a rotating frame of reference, imitating the restricted three body problem. The linear stability of periodic orbits with period one and two is analyzed with the help of stability diagrams, showing the essential parameter dependence of the residue of the periodic orbits for these examples.Comment: 22 pages, LaTex, 4 Figure

Herschel/HIFI observations of interstellar OH+ and H2O+ towards W49N: a probe of diffuse clouds with a small molecular fraction

We report the detection of absorption by interstellar hydroxyl cations and water cations, along the sight-line to the bright continuum source W49N. We have used Herschel's HIFI instrument, in dual beam switch mode, to observe the 972 GHz N = 1 - 0 transition of OH+ and the 1115 GHz 1(11) - 0(00) transition of ortho-H2O+. The resultant spectra show absorption by ortho-H2O+, and strong absorption by OH+, in foreground material at velocities in the range 0 to 70 km/s with respect to the local standard of rest. The inferred OH+/H2O+ abundance ratio ranges from ~ 3 to ~ 15, implying that the observed OH+ arises in clouds of small molecular fraction, in the 2 - 8% range. This conclusion is confirmed by the distribution of OH+ and H2O+ in Doppler velocity space, which is similar to that of atomic hydrogen, as observed by means of 21 cm absorption measurements, and dissimilar from that typical of other molecular tracers. The observed OH+/H abundance ratio of a few E-8 suggests a cosmic ray ionization rate for atomic hydrogen of (0.6 - 2.4) E-16 s-1, in good agreement with estimates inferred previously for diffuse clouds in the Galactic disk from observations of interstellar H3+ and other species.Comment: Accepted for publication in A&A Letter

Interstellar OH+, H2O+ and H3O+ along the sight-line to G10.6-0.4

We report the detection of absorption lines by the reactive ions OH+, H2O+ and H3O+ along the line of sight to the submillimeter continuum source G10.6$-$0.4 (W31C). We used the Herschel HIFI instrument in dual beam switch mode to observe the ground state rotational transitions of OH+ at 971 GHz, H2O+ at 1115 and 607 GHz, and H3O+ at 984 GHz. The resultant spectra show deep absorption over a broad velocity range that originates in the interstellar matter along the line of sight to G10.6$-$0.4 as well as in the molecular gas directly associated with that source. The OH+ spectrum reaches saturation over most velocities corresponding to the foreground gas, while the opacity of the H2O+ lines remains lower than 1 in the same velocity range, and the H3O+ line shows only weak absorption. For LSR velocities between 7 and 50 kms$^{-1}$ we estimate total column densities of $N$(OH+) $> 2.5 \times 10^{14}$ cm$^{-2}$, $N$(H2O+) $\sim 6 \times 10^{13}$ cm$^{-2}$ and $N$(H3O+) $\sim 4.0 \times 10^{13}$ cm$^{-2}$. These detections confirm the role of O$^+$ and OH$^+$ in initiating the oxygen chemistry in diffuse molecular gas and strengthen our understanding of the gas phase production of water. The high ratio of the OH+ by the H2O+ column density implies that these species predominantly trace low-density gas with a small fraction of hydrogen in molecular form