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We review recent results regarding the problem of the stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds. We shall describe techniques and methods from smooth and non-smooth geometry, the fruitful combination of which revealed particularly effective. Furthermore, we present a self-contained overview of the proof of the stability of the Sobolev inequality on manifolds with non-negative Ricci curvature and Euclidean volume growth, adopting a direct strategy tailored to this setting. Finally, we discuss related stability results and present some open problems.

Policy evaluation via Monte Carlo (MC) simulation is at the core of many MC Reinforcement Learning (RL) algorithms (e.g., policy gradient methods). In this context, the designer of the learning system specifies an interaction budget that the agent usually spends by collecting trajectories of fixed length within a simulator. However, is this data collection strategy the best option? To answer this question, in this paper, we propose as a quality index a surrogate of the mean squared error of a return estimator that uses trajectories of different lengths, i.e., \emph{truncated}. Specifically, this surrogate shows the sub-optimality of the fixed-length trajectory schedule. Furthermore, it suggests that adaptive data collection strategies that spend the available budget sequentially can allocate a larger portion of transitions in timesteps in which more accurate sampling is required to reduce the error of the final estimate. Building on these findings, we present an adaptive algorithm called Robust and Iterative Data collection strategy Optimization (RIDO). The main intuition behind RIDO is to split the available interaction budget into mini-batches. At each round, the agent determines the most convenient schedule of trajectories that minimizes an empirical and robust version of the surrogate of the estimator's error. After discussing the theoretical properties of our method, we conclude by assessing its performance across multiple domains. Our results show that RIDO can adapt its trajectory schedule toward timesteps where more sampling is required to increase the quality of the final estimation.

We study fine P\'olya-Szeg\H{o} rearrangement inequalities into weighted intervals for Sobolev functions and functions of bounded variation defined on metric measure spaces supporting an isoperimetric inequality. We then specialize this theory to spaces with synthetic Ricci lower bounds in both finite and infinite dimension and characterize equality cases under minimal assumptions. As applications of our theory, we show new results around geometric and functional inequalities under Ricci lower bounds answering also questions raised in the literature. Finally, we study further settings and deduce a Faber-Krahn theorem on Euclidean spaces with radial log-convex densities, a boosted P\'olya-Szeg\H{o} inequality with asymmetry reminder on weighted convex cones, the rigidity of Sobolev inequalities on Euclidean spaces outside a convex set and a general lower bound for Neumann eigenvalues on open sets in metric spaces.

In this paper, we investigate the presence of additive bias in Large Language Models (LLMs), drawing a parallel to the cognitive bias observed in humans where individuals tend to favor additive over subtractive changes. Using a series of controlled experiments, we tested various LLMs, including GPT-3.5 Turbo, Claude 3.5 Sonnet, Mistral, Math$\Sigma$tral, and Llama 3.1, on tasks designed to measure their propensity for additive versus subtractive modifications. Our findings demonstrate a significant preference for additive changes across all tested models. For example, in a palindrome creation task, Llama 3.1 favored adding letters 97.85% of the time over removing them. Similarly, in a Lego tower balancing task, GPT-3.5 Turbo chose to add a brick 76.38% of the time rather than remove one. In a text summarization task, Mistral 7B produced longer summaries in 59.40% to 75.10% of cases when asked to improve its own or others' writing. These results indicate that, similar to humans, LLMs exhibit a marked additive bias, which might have implications when LLMs are used on a large scale. Addittive bias might increase resource use and environmental impact, leading to higher economic costs due to overconsumption and waste. This bias should be considered in the development and application of LLMs to ensure balanced and efficient problem-solving approaches., Comment: 25 pages, 8 figures

The remnant black hole from a binary coalescence emits ringdown gravitational waves characterized by quasinormal modes, which depend solely on the remnant's mass and spin. In contrast, the ringdown amplitudes and phases are determined by the properties of the merging progenitors. Accurately modeling these amplitudes and phases reduces systematic biases in parameter estimation and enables the development and performance of rigorous tests of general relativity. We present a state-of-the-art, data-driven surrogate model for ringdown amplitudes and phases, leveraging Gaussian process regression trained against SXS numerical-relativity simulations. Focusing on nonprecessing, quasicircular binary black holes, our model offers the most comprehensive fit that includes 16 emission modes, incorporating overtones and quadratic contributions. Our surrogate model achieves reconstruction errors that are approximately 2 orders of magnitude smaller than the typical measurement errors of current gravitational-wave interferometers. An additional benefit of our approach is its flexibility, which allows for future extensions to include features such as eccentricity and precession, broadening the scope of its applicability to more generic astrophysical scenarios. Finally, we are releasing our model in a ready-to-use package called postmerger., Comment: 12 pages, 7 figures; code usage examples are available at https://github.com/cpacilio/postmerger . v2: matches published version

We study periodic tessellations of the Euclidean space with unequal cells arising from the minimization of perimeter functionals. Existence results and qualitative properties of minimizers are discussed for different classes of problems, involving local and non-local perimeters. Regularity is then addressed in the general case under volume penalization, and in the planar case with the standard perimeter, prescribing the volumes of each cell. Finally, we show the optimality of hexagonal tilings among partitions with almost equal areas., Comment: 26 Pages, 4 Figures

We study quantitative stability results for different classes of Sobolev inequalities on general compact Riemannian manifolds. We prove that, up to constants depending on the manifold, a function that nearly saturates a critical Sobolev inequality is quantitatively $W^{1,2}$-close to a non-empty set of extremal functions, provided that the corresponding optimal Sobolev constant satisfies a suitable strict bound. The case of sub-critical Sobolev inequalities is also covered. Finally, we discuss degenerate phenomena in our quantitative controls., Comment: Comments welcome!

We consider the two-dimensional Rayeigh-B\'enard convection problem between Navier-slip fixed-temperature boundary conditions and present a new upper bound for the Nusselt number. The result, based on a localization principle for the Nusselt number and an interpolation bound, exploits the regularity of the flow. On one hand our method yields a shorter proof of the celebrated result in Whitehead & Doering (2011) in the case of free-slip boundary conditions. On the other hand, its combination with a new, refined estimate for the pressure gives a substantial improvement of the interpolation bounds in Drivas et al. (2022) for slippery boundaries. A rich description of the scaling behaviour arises from our result: depending on the magnitude of the Prandtl number and slip-length, our upper bounds indicate five possible scaling laws: $\textit{Nu} \sim (L_s^{-1}\textit{Ra})^{\frac{1}{3}}$, $\textit{Nu} \sim (L_s^{-\frac{2}{5}}\textit{Ra})^{\frac{5}{13}}$, $\textit{Nu} \sim \textit{Ra}^{\frac{5}{12}}$, $\textit{Nu} \sim \textit{Pr}^{-\frac{1}{6}} (L_s^{-\frac{4}{3}}\textit{Ra})^{\frac{1}{2}}$ and $\textit{Nu} \sim \textit{Pr}^{-\frac{1}{6}} (L_s^{-\frac{1}{3}}\textit{Ra})^{\frac{1}{2}}$, Comment: 18 pages, 1 figure, 1 table

The climate change emergency calls for a reduction in energy consumption in all human activities and production processes. The radio broadcasting industry is no exception. However, reducing energy requirements by uniformly cutting the radiated power at every transmitter can potentially impair the quality of service. A careful evaluation and optimization study are in order. In this paper, by analyzing the Italian frequency modulation analog broadcasting service, we show that it is indeed possible to significantly reduce the energy consumption of the broadcasters without sacrificing the quality of the service, rather, even getting improvements.

This paper investigates the large-time behavior of a buoyancy-driven fluid without thermal diffusion under Navier-slip boundary conditions in a bounded domain with Lipschitz-continuous second derivatives. After establishing improved regularity for classical solutions, we analyze their large-time asymptotics. Specifically, we show that the solutions converge to a state where, as $t \rightarrow \infty$, $\|u\|_{W^{1,p}} \rightarrow 0$, and hydrostatic balance is achieved in the weak topology of $L^2$. Additionally, we determine the necessary conditions for the hydrostatic balance to be attained in the strong topology in the large-time limit. We then analyze a particular steady state, the hydrostatic equilibrium, characterized by $ u = 0 $, $ \theta = \beta x_2 + \gamma $, and $ p = \frac{\beta}{2}x_2^2 + \gamma x_2 + \delta $. In a periodic strip, we establish the linear stability of this state for $\beta > 0$, indicating that the temperature is vertically stably stratified. This work builds upon the results in [Doering et al.], which focus on free-slip boundary conditions, as well as recent studies [Ayd{\i}n, Kukavica, Ziane; Ayd{\i}n, Jayanti] that address no-slip boundary conditions. Notably, the novelty of this study lies in the ability to directly bound the pressure term, made possible by the Navier-slip boundary conditions., Comment: 35 pages

Cluster synchronization is of paramount importance for the normal functioning of numerous technological and natural systems. Deviations from normal cluster synchronization patterns are closely associated with various malfunctions, such as neurological disorders in the brain. Therefore, it is crucial to restore normal system functions by stabilizing the appropriate cluster synchronization patterns. Most existing studies focus on designing controllers based on state measurements to achieve system stabilization. However, in many real-world scenarios, measuring system states, such as neuronal activity in the brain, poses significant challenges, rendering the stabilization of such systems difficult. To overcome this challenge, in this paper, we employ an open-loop control strategy, vibrational control, which does not requires any state measurements. We establish some sufficient conditions under which vibrational inputs stabilize cluster synchronization. Further, we provide a tractable approach to design vibrational control. Finally, numerical experiments are conducted to demonstrate our theoretical findings., Comment: Submitted to Open Journal of Control Systems

Many natural and man-made network systems need to maintain certain patterns, such as working at equilibria or limit cycles, to function properly. Thus, the ability to stabilize such patterns is crucial. Most of the existing studies on stabilization assume that network systems states can be measured online so that feedback control strategies can be used. However, in many real-world scenarios, systems states, e.g., neuronal activity in the brain, are often difficult to measure. In this paper, we take this situation into account and study the stabilization problem of linear network systems with an open-loop control strategy (vibrational control). We derive a graph-theoretic sufficient condition for structural vibrational stabilizability, under which network systems can always be stabilized. We further provide an approach to select the locations in the network for control placement and design corresponding vibrational inputs to stabilize systems that satisfy this condition. Finally, we provide some numerical results that demonstrate the validity of our theoretical findings.

Snoring is a common disorder that affects people's social and marital lives. The annoyance caused by snoring can be partially solved with active noise control systems. In this context, the present work aims at introducing an enhanced system based on the use of a convolutional recurrent neural network for snoring activity detection and a delayless subband approach for active snoring cancellation. Thanks to several experiments conducted using real snoring signals, this work shows that the active snoring cancellation system achieves better performance when the snoring activity detection stage is turned on, demonstrating the beneficial effect of a preliminary snoring detection stage in the perspective of snoring cancellation.

We study maps of bounded variation defined on a metric measure space and valued into a metric space. Assuming the source space to satisfy a doubling and Poincar\'e property, we produce a well-behaved relaxation theory via approximation by simple maps. Moreover, several equivalent characterizations are given, including a notion in weak duality with test plans., Comment: 38 pages

Motivated by [7], we study the advection-hyperdiffusion equation in the whole space in two and three dimensions with the goal of understanding the decay in time of the $H^{-1}$- and $L^2$-norm of the solutions. We view the advection term as a perturbation of the hyperdiffusion equation and employ the Fourier-splitting method first introduced by Schonbek in [8] for scalar parabolic equations and later generalized to a broader class of equations including Navier-Stokes equations and magneto-hydrodynamic systems. This approach consists of decomposing the Fourier space along a sphere with radius decreasing in time. Combining the Fourier-splitting method with classical PDE techniques applied to the hyperdiffusion equation we find a lower bound for the $H^{-1}$-norm by interpolation., Comment: 16 pages

We consider two-dimensional Rayleigh-B\'enard convection with Navier-slip and fixed temperature boundary conditions at the two horizontal rough walls described by the height function $h$. We prove rigorous upper bounds on the Nusselt number $\text{Nu}$ which capture the dependence on the curvature of the boundary $\kappa$ and the (non-constant) friction coefficient $\alpha$ explicitly. If $h\in W^{2,\infty}$ and $\kappa$ satisfies a smallness condition with respect to $\alpha$, we find $$ \text{Nu}\lesssim \text{Ra}^{\frac{1}{2}}+\|\kappa\|_{\infty}\,,$$ where $\text{Ra}$ is the Rayleigh number, which agrees with the predicted Spiegel-Kraichnan scaling when $\kappa=0$. This bound is obtained via local regularity estimates in a small strip at the boundary. When $h\in W^{3,\infty}$, the functions $\kappa$ and $\alpha$ are sufficiently small in $L^{\infty}$ and the Prandtl number $\Pr$ is sufficiently large, we prove upper bounds using the background field method, which interpolate between $\text{Ra}^{\frac{1}{2}}$ and $\text{Ra}^{\frac{5}{12}}$ with non-trivial dependence on $\alpha$ and $\kappa$. These bounds agree with the result in Drivas et al (2022 Phil. Trans. R. Soc. A 380 20210025) for flat boundaries and constant friction coefficient. Furthermore, in the regime $\Pr\geq \text{Ra}^{\frac 57}$, we improve the $\text{Ra}^{\frac 12}$-upper bound, showing $$\text{Nu}\lesssim_{\alpha,\kappa}\text{Ra}^{\frac{3}{7}}\,,$$ where $\lesssim_{\alpha,\kappa}$ hides an additional dependency of the implicit constant on $\alpha$ and $\kappa$., Comment: 39 pages, 5 figures

We study the qualitative stability of two classes of Sobolev inequalities on Riemannian manifolds. In the case of positive Ricci curvature, we prove that an almost extremal function for the sharp Sobolev inequality is close to an extremal function of the round sphere. In the setting of non-negative Ricci curvature and Euclidean volume growth, we show an analogous result in comparison with the extremal functions in the Euclidean Sobolev inequality. As an application, we deduce a stability result for minimizing Yamabe metrics. The arguments rely on a generalized Lions' concentration compactness on varying spaces and on rigidity results of Sobolev inequalities on singular spaces., Comment: Revised version: moved some technical results to a new Appendix B, fixed Lemma 6.1, to appear in Adv. Math

Although the dynamics of colloids in the vicinity of a solid interface has been widely characterized in the past, experimental studies of Brownian diffusion close to an air-water interface are rare and limited to particle-interface gap distances larger than the particle size. At the still unexplored lower distances, the dynamics is expected to be extremely sensitive to boundary conditions at the air-water interface. There, ad hoc experiments would provide a quantitative validation of predictions. Using a specially designed dual wave interferometric setup, the 3D dynamics of 9 micrometers diameter particles at a few hundreds of nanometers from an air-water interface is here measured in thermal equilibrium. Intriguingly, while the measured dynamics parallel to the interface approaches expected predictions for slip boundary conditions, the Brownian motion normal to the interface is very close to the predictions for no-slip boundary conditions. These puzzling results are rationalized considering current models of incompressible interfacial flow and deepened developing an ad hoc model which considers the contribution of tiny concentrations of surface active particles at the interface. We argue that such condition governs the particle dynamics in a large spectrum of systems ranging from biofilm formation to flotation process., Comment: 33 pages, 5 figures. Submitted to Journal of Colloid and Interface Science