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Anticipating bifurcations of random dynamical systems through tails of stationary densities
No full textWe develop an early-warning signal for bifurcations of one-dimensional random difference equations with additive bounded noise, based on the asymptotic behaviour of the stationary density near a boundary of its support. We demonstrate the practical use in numerical examples
A General Framework for the Asymptotic Analysis of Moist Atmospheric Flows
No full textWe deal with asymptotic analysis for the derivation of partial differential equation models for geophysical flows in the Earth’s atmosphere with moist process closures, and we study their mathematical properties. Starting with the Navier–Stokes equations for dry air, we put the seminal papers of Klein, Majda et al. in a unified context and then discuss the appropriate extension to moist air. In particular, we deal with the scale-independent distinguished limit for the universal parameters of atmospheric motion for moist air, with the Clausius–Clapeyron relation that links saturation vapor pressure and air temperature, and with the mathematical formulation of phase changes associated with cloud formation and rain production. We conclude with a discussion of the precipitating quasi-geostrophic models introduced by Smith and Stechmann. Our intent is, on the one hand, to convey the problems arising at the modeling stage to mathematicians; on the other hand, we want to present the relevant mathematical methods and results to meteorologists
An Allen-Cahn tumor growth model with temperature
No full textIn this paper, we propose a new non-isothermal Allen-Cahn (Ginzburg-Landau) model for tumor growth. After deriving it using a microforces approach, we study its well-posedness. In particular, we are able to prove the existence and uniqueness of a local and global-in-time solution to our PDE system
Dynamics of systems with varying number of particles: From Liouville equations to general master equations for open systems
No full textA varying number of particles is one of the most relevant characteristics of systems of interest in nature and technology, ranging from the exchange of energy and matter with the surrounding environment to the change of particle number through internal dynamics such as reactions. The physico-mathematical modeling of these systems is extremely challenging, with the major difficulty being the time dependence of the number of degrees of freedom and the additional constraint that the increment or reduction of the number and species of particles must not violate basic physical laws. Theoretical models, in such a case, represent the key tool for the design of computational strategies for numerical studies that deliver trustful results. In this manuscript, we review complementary physico-mathematical approaches of varying number of particles inspired by rather different specific numerical goals. As a result of the analysis on the underlying common structure of these models, we propose a unifying master equation for general dynamical systems with varying number of particles. This equation embeds all the previous models and can potentially model a much larger range of complex systems, ranging from molecular to social agent-based dynamics
Time-asymptotic self-similarity of the damped compressible Euler equations in parabolic scaling variables
No full textWe study the long-time behavior of solutions to the compressible Euler equations with frictional damping in the whole space, where we prescribe direction-dependent values for the density at spatial infinity. To this end, we transform the system into parabolic scaling variables and derive a relative entropy inequality, which allows to conclude the convergence of the density towards a self-similar solution to the porous medium equation while the associated limit momentum is governed by Darcy's law. Moreover, we obtain convergence rates that explicitly depend on the flatness of the limit profile. While we focus on weak solutions in the one-dimensional case, we extend our results to energy-variational solutions in the multi-dimensional setting
Random attractors on countable state spaces
No full textWe study the synchronization behavior of discrete-time Markov chains on countable state spaces. Representing a Markov chain in terms of a random dynamical system, which describes the collective dynamics of trajectories driven by the same noise, allows for the characterization of synchronization via random attractors.
We establish the existence and uniqueness of a random attractor under mild conditions and show that forward and pullback attraction are equivalent in our setting. Additionally, we provide a sufficient condition for reaching the random attractor, or synchronization respectively, in a time of finite mean.
By introducing insulated and synchronizing sets, we structure the state space with respect to the synchronization behavior and characterize the size of the random attractor
On the conservation of helicity by weak solutions of the 3D Euler and inviscid MHD equations
No full textClassical solutions of the three-dimensional Euler equations of an ideal incompressible fluid conserve the helicity. We introduce a new weak formulation of the vorticity formulation of the Euler equations in which (by implementing the Bony paradifferential calculus) the advection terms are interpreted as paraproducts for weak solutions with low regularity. Using this approach we establish an equation of local helicity balance, which gives a rigorous foundation to the concept of local helicity density and flux at low regularity. We provide a sufficient criterion for helicity conservation which is weaker than many of the existing sufficient criteria for helicity conservation in the literature.
Subsequently, we prove a sufficient condition for the helicity to be conserved in the zero viscosity limit of the Navier–Stokes equations. Moreover, we establish a relation between the defect measure (which is part of the local helicity balance) and a third-order structure function for solutions of the Euler equations. As a byproduct of the approach introduced in this paper, we also obtain a new sufficient condition for the conservation of magnetic helicity in the inviscid MHD equations, as well as for the kinematic dynamo model.
Finally, it is known that classical solutions of the ideal (inviscid) MHD equations which have divergence-free initial data will remain divergence-free, but this need not hold for weak solutions. We show that weak solutions of the ideal MHD equations arising as weak-
limits of Leray–Hopf weak solutions of the viscous and resistive MHD equations remain divergence-free in time
A dynamic p-Laplacian
No full textWe generalise the dynamic Laplacian introduced in (Froyland, 2015) to a dynamic p-Laplacian, in analogy to the generalisation of the standard 2-Laplacian to the standard p-Laplacian for p>1. Spectral properties of the dynamic Laplacian are connected to the geometric problem of finding "coherent" sets with persistently small boundaries under dynamical evolution, and we show that the dynamic p-Laplacian shares similar geometric connections. In particular, we prove that the first eigenvalue of the dynamic p-Laplacian with Dirichlet boundary conditions exists and converges to a dynamic version of the Cheeger constant introduced in (Froyland, 2015) as p→1. We develop a numerical scheme to estimate the leading eigenfunctions of the (nonlinear) dynamic p-Laplacian, and through a series of examples we investigate the behaviour of the level sets of these eigenfunctions. These level sets define the boundaries of sets in the domain of the dynamics that remain coherent under the dynamical evolution
Filtered data based estimators for stochastic processes driven by colored noise
No full textWe consider the problem of estimating unknown parameters in stochastic differential equations driven by colored noise, which we model as a sequence of Gaussian stationary processes with decreasing correlation time. We aim to infer parameters in the limit equation, driven by white noise, given observations of the colored noise dynamics. We consider both the maximum likelihood and the stochastic gradient descent in continuous time estimators, and we propose to modify them by including filtered data. We provide a convergence analysis for our estimators showing their asymptotic unbiasedness in a general setting and asymptotic normality under a simplified scenario
FIORA: Local neighborhood-based prediction of compound mass spectra from single fragmentation events
No full textNon-targeted metabolomics holds great promise for advancing precision medicine and biomarker discovery. However, identifying compounds from tandem mass spectra remains a challenging task due to the incomplete nature of spectral reference libraries. Augmenting these libraries with simulated mass spectra can provide the necessary references to resolve unmatched spectra, but generating high-quality data is difficult. In this study, we present FIORA, an open-source graph neural network designed to simulate tandem mass spectra. Our main contribution lies in utilizing the molecular neighborhood of bonds to learn breaking patterns and derive fragment ion probabilities. FIORA not only surpasses state-of-the-art fragmentation algorithms, ICEBERG and CFM-ID, in prediction quality, but also facilitates the prediction of additional features, such as retention time and collision cross section. Utilizing GPU acceleration, FIORA enables rapid validation of putative compound annotations and large-scale expansion of spectral reference libraries with high-quality predictions