Abstract:
Large-scale persistent vortices are known to form easily in 2D disks via the Rossby wave or the baroclinic instability. In 3D, however, their formation and stability is a complex issue and still a matter of debate. We study the formation of vortices by the Rossby wave instability in a stratified inviscid disk and describe their three dimensional structure, stability and long term evolution. Numerical simulations are performed using a fully compressible hydrodynamical code based on a second order finite volume method. We assume a perfect gas law and a non-homentropic adiabatic flow.The Rossby wave instability is found to proceed in 3D in a similar way as in 2D. Vortices produced by the instability look like columns of vorticity in the whole disk thickness; the small vertical motions are related to a weak inclination of the vortex axis appearing during the development of the RWI. Vortices with aspect ratios larger than 6 are unaffected by the elliptical instability. They relax to a quasi-steady columnar structure which survives hundred of rotations while slowly migrating inward toward the star at a rate that reduces with the vortex aspect ratio. Vortices with a smaller aspect ratio are by contrast affected by the elliptic instability. Short aspect ratio vortices are completely destroyed in a few orbital periods. Vortices with an intermediate aspect ratio are partially destroyed by the elliptical instability in a region away from the mid-plane where the disk stratification is sufficiently large. Elongated Rossby vortices can survive a large number of orbital periods in protoplanetary disks in the form of vorticity columns. They could play a significant role in the evolution of the gas and the gathering of the solid particles to form planetesimals or planetary cores, a possibility that receives a renewed interest with the recent discovery of a particle trap in the disk of Oph IRS48.

Abstract:
The three-dimensional stability problem of a stretched stationary vortex is addressed in this letter. More specifically, we prove that the discrete part of the temporal spectrum is only associated with two-dimensional perturbations.

Abstract:
In game theory, the concept of Nash equilibrium reflects the collective stability of some individual strategies chosen by selfish agents. The concept pertains to different classes of games, e.g. the sequential games, where the agents play in turn. Two existing results are relevant here: first, all finite such games have a Nash equilibrium (w.r.t. some given preferences) iff all the given preferences are acyclic; second, all infinite such games have a Nash equilibrium, if they involve two agents who compete for victory and if the actual plays making a given agent win (and the opponent lose) form a quasi-Borel set. This article generalises these two results via a single result. More generally, under the axiomatic of Zermelo-Fraenkel plus the axiom of dependent choice (ZF+DC), it proves a transfer theorem for infinite sequential games: if all two-agent win-lose games that are built using a well-behaved class of sets have a Nash equilibrium, then all multi-agent multi-outcome games that are built using the same well-behaved class of sets have a Nash equilibrium, provided that the inverse relations of the agents' preferences are strictly well-founded.

Abstract:
We give examples of quasi-hyperbolic dynamical systems with the following properties : polynomial decay of correlations, convergence in law toward a non gaussian law of the ergodic sums (divided by $n^{3/4}$) associated to non degenerated regular functions.

Abstract:
A real-valued game has the finite improvement property (FIP), if starting from an arbitrary strategy profile and letting the players change strategies to increase their individual payoffs in a sequential but non-deterministic order always reaches a Nash equilibrium. E.g., potential games have the FIP. Many of them have the FIP by chance nonetheless, since modifying even a single payoff may ruin the property. This article characterises (in quadratic time) the class of the finite games where FIP not only holds but is also preserved when modifying all the occurrences of an arbitrary payoff. The characterisation relies on a pattern-matching sufficient condition for games (finite or infinite) to enjoy the FIP, and is followed by an inductive description of this class. A real-valued game is weakly acyclic if the improvement described above can reach a Nash equilibrium. This article characterises the finite such games using Markov chains and almost sure convergence to equilibrium. It also gives an inductive description of the two-player such games.

Abstract:
Subgame perfect equilibria are specific Nash equilibria in perfect information games in extensive form. They are important because they relate to the rationality of the players. They always exist in infinite games with continuous real-valued payoffs, but may fail to exist even in simple games with slightly discontinuous payoffs. This article considers only games whose outcome functions are measurable in the Hausdorff difference hierarchy of the open sets (\textit{i.e.} $\Delta^0_2$ when in the Baire space), and it characterizes the families of linear preferences such that every game using these preferences has a subgame perfect equilibrium: the preferences without infinite ascending chains (of course), and such that for all players $a$ and $b$ and outcomes $x,y,z$ we have $\neg(z <_a y <_a x \,\wedge\, x <_b z <_b y)$. Moreover at each node of the game, the equilibrium constructed for the proof is Pareto-optimal among all the outcomes occurring in the subgame. Additional results for non-linear preferences are presented.

Abstract:
We prove a central limit theorem with speed $n^{-1/2}$ for stationary processes satisfying a strong decorrelation hypothesis. The proof is a modification of the proof of a theorem of Rio. It is elementary but quite long and technical.

Abstract:
We study the existence of non-special divisors of degree $g$ and $g-1$ for algebraic function fields of genus $g\geq 1$ defined over a finite field $\F_q$. In particular, we prove that there always exists an effective non-special divisor of degree $g\geq 2$ if $q\geq 3$ and that there always exists a non-special divisor of degree $g-1\geq 1$ if $q\geq 4$. We use our results to improve upper and upper asymptotic bounds on the bilinear complexity of the multiplication in any extension $\F_{q^n}$ of $\F_q$, when $q=2^r\geq 16$.

Abstract:
Given an infinitesimal perturbation of a discrete-time finite Markov chain, we seek the states that are stable despite the perturbation, \textit{i.e.} the states whose weights in the stationary distributions can be bounded away from $0$ as the noise fades away. Chemists, economists, and computer scientists have been studying irreducible perturbations built with exponential maps. Under these assumptions, Young proved the existence of and computed the stable states in cubic time. We fully drop these assumptions, generalize Young's technique, and show that stability is decidable as long as $f\in O(g)$ is. Furthermore, if the perturbation maps (and their multiplications) satisfy $f\in O(g)$ or $g\in O(f)$, we prove the existence of and compute the stable states and the metastable dynamics at all time scales where some states vanish. Conversely, if the big-$O$ assumption does not hold, we build a perturbation with these maps and no stable state. Our algorithm also runs in cubic time despite the general assumptions and the additional work. Proving the correctness of the algorithm relies on new or rephrased results in Markov chain theory, and on algebraic abstractions thereof.

Abstract:
We consider the degrees of non-computability (Weihrauch degrees) of finding winning strategies (or more generally, Nash equilibria) in infinite sequential games with certain winning sets (or more generally, outcome sets). In particular, we show that as the complexity of the winning sets increases in the difference hierarchy, the complexity of constructing winning strategies increases in the effective Borel hierarchy.