Nonautonomous attractors and Young measures

Motivated by the problem of identifying a mathematical framework for the formal definition of concepts such as weather, climate and connections between them, we discuss a question of convergence of short-time time averages for random nonautonomous dynamical systems depending on a parameter. The problem is formulated by means of Young measures. Using the notion of pull-back attractor, we prove a general theorem giving a sufficient condition for the tightness of the law of the approximating problems. In a specific example, we show that the theorem applies and we characterize the unique limit point.

Analog pile-up circuit technique using a single capacitor for the readout of Skipper-CCD detectors

With Skipper-CCD detectors it is possible to take multiple samples of the charge packet collected on each pixel. After averaging the samples, the noise can be extremely reduced allowing the exact counting of electrons per pixel. In this work we present an analog circuit that, with a minimum number of components, applies a double slope integration (DSI) and at the same time averages the multiple samples, producing at its output the pixel value with sub-electron noise. For this purpose, we introduce the technique of using the DSI integrator capacitor to add the skipper samples. An experimental verification using discrete components is presented, together with an analysis of its noise sources and limitations. After averaging 400 samples it was possible to reach a readout noise of 0.18 e- rms/pix, comparable to other available readout systems. Due to its simplicity and significant reduction of the sampling requirements, this circuit technique is of particular interest in particle experiments and cameras with a high density of Skipper-CCDs.

Ergodic equilibration of Rényi entropies and replica wormholes

We study the behavior of Rényi entropies for pure states from standard assumptions about chaos in the high-energy spectrum of the Hamiltonian of a many-body quantum system. We compute the exact long-time averages of Rényi entropies and show that the quantum noise around these values is exponentially suppressed in the microcanonical entropy. For delocalized states over the microcanonical band, the long-time average approximately reproduces the equilibration proposal of H. Liu and S. Vardhan, with extra structure arising at the order of non-planar permutations. We analyze the equilibrium approximation for AdS/CFT systems describing black holes in equilibrium in a box. We extend our analysis to the situation of an evaporating black hole, and comment on the possible gravitational description of the new terms in our approximation.

Inviscid limit for the damped generalized incompressible Navier-Stokes equations on $ \mathbb{T}^2 $

<p style='text-indent:20px;'>In this paper, for the damped generalized incompressible Navier-Stokes equations on <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{T}^{2} $\end{document}</tex-math></inline-formula> as the index <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> of the general dissipative operator <inline-formula><tex-math id="M4">\begin{document}$ (-\Delta)^{\alpha} $\end{document}</tex-math></inline-formula> belongs to <inline-formula><tex-math id="M5">\begin{document}$ (0,\frac{1}{2}] $\end{document}</tex-math></inline-formula>, we prove the absence of anomalous dissipation of the long time averages of entropy. We also give a note to show that, by using the <inline-formula><tex-math id="M6">\begin{document}$ L^{\infty} $\end{document}</tex-math></inline-formula> bounds given in Caffarelli et al. [<xref ref-type="bibr" rid="b4">4</xref>], the absence of anomalous dissipation of the long time averages of energy for the forced SQG equations established in Constantin et al. [<xref ref-type="bibr" rid="b12">12</xref>] still holds under a slightly weaker conditions <inline-formula><tex-math id="M7">\begin{document}$ \theta_{0}\in L^{1}(\mathbb{R}^{2})\cap L^{2}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ f \in L^{1}(\mathbb{R}^{2})\cap L^{p}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula> with some <inline-formula><tex-math id="M9">\begin{document}$ p&gt;2 $\end{document}</tex-math></inline-formula>.</p>

Exponential turnpike property for fractional parabolic equations with non-zero exterior data

We consider averages convergence as the time-horizon goes to infinity of optimal solutions of time-dependent optimal control problems to optimal solutions of the corresponding stationary optimal control problems. Assuming that the controlled dynamics under consideration are stabilizable towards a stationary solution, the following natural question arises: Do time averages of optimal controls and trajectories converge to the stationary optimal controls and states as the time-horizon goes to infinity? This question is very closely related to the so-called turnpike property that shows that, often times, the optimal trajectory joining two points that are far apart, consists in, departing from the point of origin, rapidly getting close to the steady-state (the turnpike) to stay there most of the time, to quit it only very close to the final destination and time. In the present paper we deal with heat equations with non-zero exterior conditions (Dirichlet and nonlocal Robin) associated with the fractional Laplace operator $(-\Delta)^s$ ( $0<s<1$ ). We prove the turnpike property for the nonlocal Robin optimal control problem and the exponential turnpike property for both Dirichlet and nonlocal Robin optimal control problems.