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An almost tight lower bound for plurality consensus with undecided state dynamics in the population protocol model
No full textWe revisit the majority problem in the population protocol communication model, as first studied by Angluin et al. (Distributed Computing 2008). We consider a more general version of this problem known as plurality consensus, which has already been studied intensively in the literature. In this problem, each node in a system of n nodes, has initially one of k different opinions, and they need to agree on the (relative) majority opinion. In particular, we consider the important and intensively studied model of Undecided State Dynamics.
Our main contribution is an almost tight lower bound on the stabilization time: we prove that there exists an initial configuration, even with bias \Delta = \omega(\sqrt{n\log n}), where stabilization requires \Omega(kn\log \frac {\sqrt n} {k \log n}) interactions, or equivalently, \Omega(k\log \frac {\sqrt n} {k \log n}) parallel time for any k = o\left(\frac {\sqrt n}{\log n}\right). This bound is tight for any k \le n^{\frac 1 2 - \epsilon}, where \epsilon >0 can be any small constant, as Amir et al.~(PODC'23) gave a O(k\log n) parallel time upper bound for k = O\left(\frac {\sqrt n} {\log ^2 n}\right)
When is liquid democracy possible?: On the manipulation of variance
No full textLiquid democracy is a transitive vote delegation mechanism over voting graphs. It enables each voter to delegate their vote(s) to another better-informed voter, with the goal of collectively making a better decision. The question of whether liquid democracy outperforms direct voting has been previously studied in the context of local delegation mechanisms (where voters can only delegate to someone in their neighbourhood) and binary decision problems. It has previously been shown that it is impossible for local delegation mechanisms to outperform direct voting in general graphs. This raises the question: for which classes of graphs do local delegation mechanisms yield good results?
In this work, we analyse (1) properties of specific graphs and (2) properties of local delegation mechanisms on these graphs, determining where local delegation actually outperforms direct voting. We show that a critical graph property enabling liquid democracy is that the voting outcome of local delegation mechanisms preserves a sufficient amount of variance, thereby avoiding situations where delegation falls behind direct voting1. These insights allow us to prove our main results, namely that there exist local delegation mechanisms that perform no worse and in fact quantitatively better than direct voting in natural graph topologies like complete, random d-regular, and bounded degree graphs, lending a more nuanced perspective to previous impossibility results
L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model
No full textWe prove upper bounds on the L∞-Wasserstein distance from optimal transport between strongly log-concave probability densities and log-Lipschitz perturbations. In the simplest setting, such a bound amounts to a transport-information inequality involving the L∞-Wasserstein metric and the relative L∞-Fisher information. We show that this inequality can be sharpened significantly in situations where the involved densities are anisotropic. Our proof is based on probabilistic techniques using Langevin dynamics. As an application of these results, we obtain sharp exponential rates of convergence in Fisher’s infinitesimal model from quantitative genetics, generalising recent results by Calvez, Poyato, and Santambrogio in dimension 1 to arbitrary dimensions
A duality between surface charge and work function in scanning Kelvin probe microscopy
No full textScanning Kelvin probe microscopy (SKPM) is a powerful technique for macroscopic imaging of the electrostatic potential above a surface. Though most often used to image work-function variations of conductive surfaces, it can also be used to probe the surface charge on insulating surfaces. In both cases, relating the measured potential to the underlying signal is non-trivial. Here, general relationships are derived between the measured SKPM voltage and the underlying source, revealing either can be cast as a convolution with an appropriately scaled point spread function (PSF). For charge that exists on a thin insulating layer above a conductor, the PSF has the same shape as what would occur from a work-function variation alone, differing by a simple scaling factor. This relationship is confirmed by: (1) backing it out from finite-element simulations of work-function and charge signals, and (2) experimentally comparing the measured PSF from a small work-function target to that from a small charge spot. This scaling factor is further validated by comparing SKPM charge measurements with Faraday cup measurements for highly charged samples from contact-charging experiments. These results highlight a heretofore unappreciated connection between SKPM voltage and charge signals, offering a rigorous recipe to extract either from experimental data
A high fraction of close massive binary stars at low metallicity
No full textAt high metallicity, a majority of massive stars have at least one close stellar companion. The evolution of such binaries is subject to strong interaction processes, which heavily impact the characteristics of their life-ending supernova and compact remnants. For the low-metallicity environments of high-redshift galaxies, constraints on the multiplicity properties of massive stars over the separation range leading to binary interaction are crucially missing. Here we show that the presence of massive stars in close binaries is ubiquitous, even at low metallicity. Using the Very Large Telescope, we obtained multi-epoch radial velocity measurements of a representative sample of 139 massive O-type stars across the Small Magellanic Cloud, which has a metal content of about one-fifth of the solar value. We find that 45% of them show radial velocity variations that demonstrate that they are members of close binary systems, and predominantly have orbital periods shorter than 1 year. Correcting for observational biases indicates that at least 70+11−6 % of the O stars in our sample are in close binaries, and that at least 68+7
−8% of all O stars interact with a companion star during their lifetime. We found no evidence supporting a statistically significant trend of the multiplicity properties with metallicity. Our results indicate that multiplicity and binary interactions govern the evolution of massive stars and determine their cosmic feedback and explosive fates
Bumps on the road: The way to clean relaxation dispersion magic-angle spinning NMR
No full textMicrosecond-to-millisecond motions are instrumental for many biomolecular functions, including enzymatic activity and ligand binding. Bloch-McConnell Relaxation Dispersion (BMRD) Nuclear Magnetic Resonance (NMR) spectroscopy is a key technique for studying these dynamic processes. While BMRD experiments are routinely used to probe protein motions in solution, the experiment is more demanding in the solid state, where dipolar couplings complicate the spin dynamics. It is believed that high deuteration levels are required and sufficient to obtain accurate and quantitative data. Here we show that even under fast magic-angle spinning and high levels of deuteration artifactual “bumps” in 15N R1ρ BMRD profiles are common. The origin of these artifacts is identified as a second-order three-spin Mixed Rotational and Rotary Resonance (MIRROR) recoupling condition. These artifacts are found to be a significant confounding factor for the accurate quantification of microsecond protein dynamics using BMRD in the solid state. We show that the application of low-power continuous wave (CW) decoupling simultaneously with the 15N spin-lock leads to the suppression of these conditions and enables quantitative measurements of microsecond exchange in the solid state. Remarkably, the application of decoupling allows the measurement of accurate BMRD even in fully protonated proteins at 100 kHz MAS, thus extending the scope of μs dynamics measurements in MAS NMR
Enhanced superconductivity at a corner for the linear BCS equation
No full textWe consider the critical temperature for superconductivity, defined via the linear BCS equation. We prove that at weak coupling the critical temperature for a sample confined to a quadrant in two dimensions is strictly larger than the one for a half-space, which in turn is strictly larger than the one for R^2. Furthermore, we prove that the relative difference of the critical temperatures vanishes in the weak coupling limit
Intensity oscillations of tropical cyclones: Surface versus mid and upper tropospheric processes
No full textSome of the classical models of tropical cyclone intensification predict tropical cyclones to intensify up to a steady intensity, which depends on surface fluxes only, without any relevant role played by convective motions in the troposphere, typically assumed to have a moist adiabatic lapse rate. Simulations performed using the non-hydrostatic, high-resolution model System for Atmosphere Modeling in idealized settings (rotating radiative-convective equilibrium on a doubly periodic domain) show early intensification consistent with these theoretical expectations, but different intensity evolution, with the cyclone undergoing an oscillation in wind speed. This oscillation can be linked to feedbacks between the cyclone intensity and air buoyancy: convective heating, radiative heating, and mixing with warm low stratospheric air warm the mid and upper troposphere of the cyclone stabilizing the air column and thus reducing its intensity. After the intensity decay phase, mid and upper tropospheric cooling, mostly through cold advection from the surroundings, cooled by radiation, rebuilds Convective Available Potential Energy, that peaks just before a new intensification phase. These idealized simulations thus highlight the potentially important interactions between a tropical cyclone, its environment and radiation
The stochastic primitive equations with nonisothermal turbulent pressure
No full textIn this paper, we introduce and study the primitive equations with non-isothermal turbulent pressure and transport noise. They are derived from the Navier–Stokes equations by employing stochastic versions of the Boussinesq and the hydrostatic approximations. The temperature dependence of the turbulent pressure can be seen as a consequence of an additive noise acting on the small vertical dynamics. For such a model we prove global well-posedness in H^1 where the noise is considered in both the Itô and Stratonovich formulations. Compared to previous variants of the primitive equations, the one considered here presents a more intricate coupling between the velocity field and the temperature. The corresponding analysis is seriously more involved than in the deterministic setting. Finally, the continuous dependence on the initial data and the energy estimates proven here are new, even in the case of isothermal turbulent pressure
ISTA Thesis
No full textThe overarching goal of this thesis is to break down the complexity of turbulent flows in terms of enumerable, coherent structures and patterns. In a five-paper series, we adopt a variety of perspectives and techniques to relate the properties of systems of increasing complexity to their underlying coherent structures.
Initially, we take a dynamical systems point of view, seeing turbulent flow as a chaotic trajectory bouncing between exact unstable solutions of the underlying equations of motion. Using persistent homology, the main tool of topological data analysis capturing the persistence across scales of topological features in a point cloud, we introduce a method that quantifies visits of turbulent trajectories to unstable time-periodic solutions, also called periodic orbits. We demonstrate this method first in the Rössler and Kuramoto–Sivashinsky systems. Using this method in 3D Kolmogorov flow, we extract a Markov chain from turbulent data, where each node corresponds to the neighbourhood of a periodic orbit. The invariant distribution of this Markov chain reproduces expectation values on turbulent data when it is used to weight averages on the respective periodic orbits.
In more realistic, wall-bounded settings, such as plane-Couette flow (pcf) driven by the relative motion of the walls, or plane-Poiseuille flow (ppf) driven by a pressure gradient, finding exact solutions is difficult. We use dynamic mode decomposition (DMD), a dimensionality reduction method for sequential data, to identify and approximate low-dimensional dynamics without knowing any exact solutions. Most spatially-extended systems are equivariant under translations, and in such cases spatial drifts dominate DMD, hindering its use in the search for and modelling of low-dimensional dynamics. We augment DMD with a symmetry reduction method trained on turbulent data to stop it from seeing translations as a feature, improving its ability to extract dynamical information in translation-equivariant systems. We find segments of turbulent trajectories that linearize well with their symmetry-reduced DMD spectra, akin to dynamics near exact solutions. Searching for harmonics in the spectra gives leads for periodic orbits with spatial drifts, one of which converges to a new solution.
In larger domains, turbulence can localize and coexist with surrounding laminar flow. Our preceding approaches are global, taking all of a domain into account at once, and cannot readily treat each localized patch individually. Working first in a minimal oblique domain that can host a single 1D-localized turbulent patch, we find that turbulence in ppf is connected to a stable periodic orbit at a flow velocity much lower than when turbulence is first onset. We show that, well in advance of sustained turbulence, chaos sets in explosively, and for long time horizons, time series are consistent with that of a random process.
Finally, in much larger domains, we study and compare 2D-localized turbulence that appears as large-scale inclined structures, called stripes, in ppf and pcf. While appearing similar, we find that stripes in these two settings differ significantly in terms of how they sustain themselves, and in higher velocities, how they proliferate