A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems

In this paper we introduce a new kind of Lax-Oleinik type operator with parameters associated with positive definite Lagrangian systems for both the time-periodic case and the time-independent case. On one hand, the new family of Lax-Oleinik type operators with an arbitrary $u\in C(M,\mathbb{R}^1)$ as initial condition converges to a backward weak KAM solution in the time-periodic case, while it was shown by Fathi and Mather that there is no such convergence of the Lax-Oleinik semigroup. On the other hand, the new family of Lax-Oleinik type operators with an arbitrary $u\in C(M,\mathbb{R}^1)$ as initial condition converges to a backward weak KAM solution faster than the Lax-Oleinik semigroup in the time-independent case.Comment: We give a new definition of Lax-Oleinik type operator; add some reference

Weak KAM for commuting Hamiltonians

For two commuting Tonelli Hamiltonians, we recover the commutation of the Lax-Oleinik semi-groups, a result of Barles and Tourin ([BT01]), using a direct geometrical method (Stoke's theorem). We also obtain a "generalization" of a theorem of Maderna ([Mad02]). More precisely, we prove that if the phase space is the cotangent of a compact manifold then the weak KAM solutions (or viscosity solutions of the critical stationary Hamilton-Jacobi equation) for G and for H are the same. As a corrolary we obtain the equality of the Aubry sets, of the Peierls barrier and of flat parts of Mather's $\alpha$ functions. This is also related to works of Sorrentino ([Sor09]) and Bernard ([Ber07b]).Comment: 23 pages, accepted for publication in NonLinearity (january 29th 2010). Minor corrections, fifth part added on Mather's $\alpha$ function (or effective Hamiltonian

Using Reinforcement Learning to Simplify Mealtime Insulin Dosing for People with Type 1 Diabetes: In-Silico Experiments

People with type 1 diabetes (T1D) struggle to calculate the optimal insulin dose at mealtime, especially when under multiple daily injections (MDI) therapy. Effectively, they will not always perform rigorous and precise calculations, but occasionally, they might rely on intuition and previous experience. Reinforcement learning (RL) has shown outstanding results in outperforming humans on tasks requiring intuition and learning from experience. In this work, we propose an RL agent that recommends the optimal meal-accompanying insulin dose corresponding to a qualitative meal (QM) strategy that does not require precise carbohydrate counting (CC) (e.g., a usual meal at noon.). The agent is trained using the soft actor-critic approach and comprises long short-term memory (LSTM) neurons. For training, eighty virtual subjects (VS) of the FDA-accepted UVA/Padova T1D adult population were simulated using MDI therapy and QM strategy. For validation, the remaining twenty VS were examined in 26-week scenarios, including intra- and inter-day variabilities in glucose. \textit{In-silico} results showed that the proposed RL approach outperforms a baseline run-to-run approach and can replace the standard CC approach. Specifically, after 26 weeks, the time-in-range ($70-180$mg/dL) and time-in-hypoglycemia ($<70$mg/dL) were $73.1\pm11.6$% and $2.0\pm 1.8$% using the RL-optimized QM strategy compared to $70.6\pm14.8$% and $1.5\pm 1.5$% using CC. Such an approach can simplify diabetes treatment, resulting in improved quality of life and glycemic outcomes.Comment: 6 pages, 4 figures, conferenc

Topological Entropy of Braids on the Torus

A fast method is presented for computing the topological entropy of braids on the torus. This work is motivated by the need to analyze large braids when studying two-dimensional flows via the braiding of a large number of particle trajectories. Our approach is a generalization of Moussafir's technique for braids on the sphere. Previous methods for computing topological entropies include the Bestvina--Handel train-track algorithm and matrix representations of the braid group. However, the Bestvina--Handel algorithm quickly becomes computationally intractable for large braid words, and matrix methods give only lower bounds, which are often poor for large braids. Our method is computationally fast and appears to give exponential convergence towards the exact entropy. As an illustration we apply our approach to the braiding of both periodic and aperiodic trajectories in the sine flow. The efficiency of the method allows us to explore how much extra information about flow entropy is encoded in the braid as the number of trajectories becomes large.Comment: 19 pages, 44 figures. SIAM journal styl

Neutral gas in Lyman-alpha emitting galaxies Haro 11 and ESO 338-IG04 measured through sodium absorption

Context. The Lyman alpha emission line of galaxies is an important tool for finding galaxies at high redshift, and thus probe the structure of the early universe. However, the resonance nature of the line and its sensitivity to dust and neutral gas is still not fully understood. Aims. We present measurements of the velocity, covering fraction and optical depth of neutral gas in front of two well known local blue compact galaxies that show Lyman alpha in emission: ESO 338-IG 04 and Haro 11. We thus test observationally the hypothesis that Lyman alpha can escape through neutral gas by being Doppler shifted out of resonance. Methods. We present integral field spectroscopy from the GIRAFFE/Argus spectrograph at VLT/FLAMES in Paranal, Chile. The excellent wavelength resolution allows us to accurately measure the velocity of the ionized and neutral gas through the H-alpha emission and Na D absorption, which traces the ionized medium and cold interstellar gas, respectively. We also present independent measurements with the VLT/X-shooter spectrograph which confirm our results. Results. For ESO 338-IG04, we measure no significant shift of neutral gas. The best fit velocity is -15 (16) km/s. For Haro 11, we see an outflow from knot B at 44 (13) km/s and infalling gas towards knot C with 32 (12) km/s. Based on the relative strength of the Na D absorption lines, we estimate low covering fractions of neutral gas (down to 10%) in all three cases. Conclusions. The Na D absorption likely occurs in dense clumps with higher column densities than where the bulk of the Ly-alpha scattering takes place. Still, we find no strong correlation between outflowing neutral gas and a high Lyman alpha escape fraction. The Lyman alpha photons from these two galaxies are therefore likely escaping due to a low column density and/or covering fraction.Comment: 9 pages, 3 figure

Spherical Solutions due to the Exterior Geometry of a Charged Weyl Black Hole

Firstly we derive peculiar spherical Weyl solutions, using a general spherically symmetric metric due to a massive charged object with definite mass and radius. Afterwards, we present new analytical solutions for relevant cosmological terms, which appear in the metrics. Connecting the metrics to a new geometric definition of a charged Black Hole, we numerically investigate the effective potentials of the total dynamical system, considering massive and massless test particles, moving on such Black Holes.Comment: 8 pages, 5 figure

The central region of M83: Massive star formation, kinematics, and the location and origin of the nucleus

We report new near-IR integral field spectroscopy of the central starburst region of the barred spiral galaxy M83 obtained with CIRPASS on Gemini-S, which we analyse in conjunction with GHaFaS Fabry-Perot data, an AAT IRIS2 Ks-band image, and near- and mid-IR imaging from the Hubble and Spitzer space telescopes. The bulk of the current star formation activity is hidden from optical view by dust extinction, but is seen in the near- and mid-IR to the north of the nucleus. This region is being fed by inflow of gas through the bar of M83, traced by the prominent dust lane entering into the circumnuclear region from the north. An analysis of stellar ages confirms that the youngest stars are indeed in the northwest. A gradual age gradient, with older stars further to the south, characterises the well-known star-forming arc in the central region of M83. Detailed analyses of the Pa beta ionised gas kinematics and near-IR imaging confirm that the kinematic centre coincides with the photometric centre of M83, and that these are offset significantly, by about 3 arcsec or 60 pc, from the visible nucleus of the galaxy. We discuss two possible options, the first of which postulates that the kinematic and photometric centre traces a galaxy nucleus hidden by a substantial amount of dust extinction, in the range A_V=3-10 mag. By combining this information with kinematic results and using arguments from the literature, we conclude that such a scenario is, however, unlikely, as is the existence of other "hidden" nuclei in M83. We thus concur with recent authors and favour a second option, in which the nucleus of the galaxy is offset from its kinematic and photometric centre. This is presumably a result of some past interaction, possibly related to the event which lies at the origin of the disturbance of the outer disk of the galaxy. (Abridged)Comment: MNRAS, in press; 16 pages latex, 15 figure

Aubry sets vs Mather sets in two degrees of freedom

We study autonomous Tonelli Lagrangians on closed surfaces. We aim to clarify the relationship between the Aubry set and the Mather set, when the latter consists of periodic orbits which are not fixed points. Our main result says that in that case the Aubry set and the Mather set almost always coincide.Comment: Revised and expanded version. New proof of Lemma 2.3 (formerly Lemma 14