Simone Panzeri’s Post

STRUCTURE-PRESERVING LIE POISSON TIME INTEGRATION Geometric derivation and structure-preserving simulation of quasi-geostrophy on the sphere Erwin Luesink, Arnout Franken, Sagy Ephrati, Bernard Geurts https://lnkd.in/e54ZzAJ2 We present a geometric derivation of the quasi-geostrophic equations on the sphere, starting from the rotating shallow water equations. We utilise perturbation series methods in vorticity and divergence variables. The derivation employs asymptotic analysis techniques, leading to a global quasi-geostrophic potential vorticity model on the sphere without approximation of the Coriolis parameter. The resulting model forms a closed system for the evolution of potential vorticity with a rich mathematical structure, including Lagrangian and Hamiltonian descriptions. Formulated using the Lie-Poisson bracket reveals the geometric invariants of the quasi-geostrophic model. Motivated by these geometric results, simulations of quasi-geostrophic flow on the sphere are presented based on structure-preserving Lie-Poisson time-integration. We explicitly demonstrate the preservation of Casimir invariants and show that the hyperbolic quasi-geostrophic equations can be simulated in a stable manner over long time. We show the emergence of longitudonal jets, wrapped around the circumference of the sphere in a general direction that is perpendicular to the axis of rotation.

Imaginary numbers aren't imaginary at all. If you don't agree, I'll argue that real numbers also exist only in our imagination! However, putting aside this naming issue, one may ask, why do we need complex numbers? Rational numbers form a field under addition and multiplication, but they aren't complete. A bounded subset of rational numbers may not have an infimum or supremum, and a Cauchy sequence of rational numbers may not converge to another rational number. Thus, ℚ is not complete, it has holes in it, which we fill by extending it to ℝ. However, real numbers aren't algebraically closed, which is why we have ℂ. If you want to provide a field structure to ℝⁿ (which is a vector space), it is only possible for n=1,2 (the closest structures are Quaternions, but they don't commute under multiplication). The pathologies that arise in real numbers, such as differentiable but not analytic functions or functions that are continuous everywhere but differentiable nowhere, do not appear in complex analysis. Every linear operator of a finite-dimensional vector space always has an eigenvalue if the field is complex (which is not true for the real fields in general). This is why we use complex vector spaces in Quantum Mechanics! Without 𝘪, the Schrödinger Equation would just be a heat equation! [ Attached photo is the famous Euler's Identity, which is, in my opinion, the most beautiful expression. ]

📃Scientific paper: Zero-Age Horizontal Branch Models for -2.5 <= [Fe/H] <= -0.5 and Their Implications for the Apparent Distance Moduli of Globular Clusters Abstract: Grids of zero-age horizontal branch (ZAHB) models are presented, along with a suitable interpolation code, for -2.5 <= [Fe/H] <= -0.5, in steps of 0.2 dex, assuming Y = 0.25 and 0.29, [O/Fe] = +0.4 and +0.6, and [m/Fe] = 0.4 for all of the other alpha elements. The HB populations of 37 globular clusters (GCs) are fitted to these ZAHBs to derive their apparent distance moduli, (m-M)_V. With few exceptions, the best estimates of their reddenings from dust maps are adopted. The distance moduli are constrained using the prediction that (M_F606W-M_F814W)_0 colours of metal-poor, main-sequence stars at M_F606W >~ 5.0 have very little sensitivity to [Fe/H]. Intrinsic (M_F336W-M_F606W)_0 colours of blue HB stars, which provide valuable connections between GCs with exclusively blue HBs and other clusters of similar metallicity that also have red HB components, limit the uncertainties of relative (m-M)_V values to within +/- 0.03-0.04 mag. The ZAHB-based distances agree quite well with the distances derived by Baumgardt & Vasiliev (2021, MNRAS, 505, 5957). Their implications for GC ages are briefly discussed. Stellar rotation and mass loss appear to be more important than helium abundance variations in explaining the colour-magnitude diagrams of second-parameter GCs (those with anomalously very blue HBs for their metallicities). ;Comment: Accepted for publication in the MNRAS; 26 page article with 20 figures Continued on ES/IODE ➡️ https://etcse.fr/Cjw01 ------- If you find this interesting, feel free to follow, comment and share. We need your help to enhance our visibility, so that our platform continues to serve you.

This is my last paper on singularities of spatial mechanisms. It is published in OA so it can be freely downloaded from the below reported link. Abstract: Multi-degree-of-freedom (multi-DOF) mechanisms generate single-DOF mechanisms by locking all their generalized coordinates but one. Here, the superposition principle is used to state a relationship between spatial multi-DOF mechanisms’ instantaneous kinematics (IK) and the IK of the single-DOF mechanisms they generate. Firstly, the relationship between the instantaneous screw axes (ISAs) of a multi-DOF mechanism and the ISAs of the single-DOF mechanisms, it generates, is found; then, it is used for its singularity analysis. In particular, the IK model of a generic multi-DOF spatial mechanism is written through the ISA locations and, successively, it is studied to identify all the singular configurations of the multi-DOF mechanism through the analysis of the single-DOF mechanisms it generates. The results are a technique for the determination of ISAs’ locations in multi-DOF spatial mechanisms and a singularity-analysis technique, for the same mechanism types, based on the singularity analysis of single-DOF spatial mechanisms. Eventually, the proposed techniques are applied to a case study. As far as this author is aware, both these results are presented for the first time in the literature.

Old stochastic calculations never die "Singularities in d-dimensional Langevin equations with anisotropic multiplicative noise and lack of self-adjointness in the corresponding Schrodinger equation" In this paper we analyse d-dimensional Langevin equations in Ito representation characterised by anisotropic multiplicative noise, composed by the superposition of an isotropic tensorial component and a radial one, and a radial power law drift term. This class of model is relevant in many contexts ranging from vortex stochastic dynamics, passive scalar transport in fully developed turbulence and second order phase transitions from active to absorbing states. The focus of the paper is the system behaviour around the singularity at vanishing distance depending on the model parameters. This can vary from regular boundary to naturally repulsive or attractive ones. The work develops in the following steps: (i) introducing a mapping that disentangle the radial dynamics from the angular one, with the first characterised by additive noise and either logarithmic or power law potential, and the second one being simply a free isotropic Brownian motion on the unitary sphere surface; (ii) applying the Feller's - Van Kampen's classification of singularities for continuous Markov processes; (iii) developing an Exponent Hunter Method to find the small distance scaling behaviour of the solutions; (iv) building a mapping into a well studied Schrodinger equation with singular potential in order to build a bridge between Feller's theory of singular boundaries of a continuous Markov process and the problem of self-adjointness of the Hamiltonian operator in quantum theory. https://lnkd.in/d3z5JasB

Hi everyone, Our paper titled "Which shapes can appear in a Curve Shortening Flow Singularity" has been posted on arxiv.org, (https://lnkd.in/esaaZXaN). The project was supervised by Prof. Sigurd Angenent, UW-Madison. The paper studies the possible tangles that can occur in singularities of solutions to plane Curve Shortening Flow. We introduce the so-called n-loop curves, which generalize Matt Grayson's Figure-Eight curve, and we conjecture a generalization of the Coiculescu-Schwarz asymptotic bow-tie result. In particular, we exhibit solutions in which more complicated tangles with more than one self-intersection disappear into a singular point. Any comments are welcome. #mathematics #Differential #geometry #AnalysisofPDE

Entropy corrected geometric Brownian motion “The geometric Brownian motion (GBM) is widely employed for modeling stochastic processes, yet its solutions are characterized by the log-normal distribution. This comprises predictive capabilities of GBM mainly in terms of forecasting applications. Here, entropy corrections to GBM are proposed to go beyond log-normality restrictions and better account for intricacies of real systems. It is shown that GBM solutions can be effectively refined by arguing that entropy is reduced when deterministic content of considered data increases. Notable improvements over conventional GBM are observed for several cases of non-log-normal distributions, ranging from a dice roll experiment to real world data” Link to article: https://lnkd.in/g2d6YTUv Read more like this: https://lnkd.in/gcMv7vXh #trading #finance

Please have a look at this manuscript 👇 it introduces a really interesting technique for data augmentation to improve the training of #datadriven #ROMs.

Extracting high-order cosmological information in galaxy surveys with power spectra -Abstract :"The reconstruction method was proposed more than a decade ago to boost the signal of baryonic acoustic oscillations measured in galaxy redshift surveys, which is one of key probes for dark energy. After moving the observed overdensities in galaxy surveys back to their initial position, the reconstructed density field is closer to a linear Gaussian field, with higher-order information moved back into the power spectrum. We find that by jointly analysing power spectra measured from the pre- and post-reconstructed galaxy samples, higher-order information beyond the 2-point power spectrum can be efficiently extracted, which generally yields an information gain upon the analysis using the pre- or post-reconstructed galaxy sample alone. This opens a window to easily use higher-order information when constraining cosmological models. " https://lnkd.in/eCRj_i7w

#article Kink, P. A Spectral Method Approach to Quadratic Normal Volatility Diffusions. Symmetry 2023, 15, 1474. 📑 Free full text: https://lnkd.in/gDZKpmfj 💡 Keywords: stochastic processes; stochastic differential equations; Itô calculus; martingales with continuous parameter; quadratic volatility #stochastic #SDEs #calculus