Explicit estimates on the measure of primary KAM tori

From KAM Theory it follows that the measure of phase points which do not lie on Diophantine, Lagrangian, "primary" tori in a nearly--integrable, real--analytic Hamiltonian system is $O(\sqrt{\varepsilon})$, if $\varepsilon$ is the size of the perturbation. In this paper we discuss how the constant in front of $\sqrt{\varepsilon}$ depends on the unperturbed system and in particular on the phase--space domain

The spin-orbit resonances of the Solar system: A mathematical treatment matching physical data

In the mathematical framework of a restricted, slightly dissipative spin-orbit model, we prove the existence of periodic orbits for astronomical parameter values corresponding to all satellites of the Solar system observed in exact spin-orbit resonance

Global properties of generic real-analytic nearly-integrable Hamiltonian systems

We introduce a new class $\mathbb{G}^n_s$ of generic real analytic potentials on $\mathbb{T}^n$ and study global analytic properties of natural nearly-integrable Hamiltonians $\frac12 |y|^2+\varepsilon f(x)$, with potential $f\in \mathbb{G}^n_s$, on the phase space $\varepsilon = B \times \mathbb{T}^n$ with $B$ a given ball in $\mathbb{R}^n$. The phase space $\mathcal{M}$ can be covered by three sets: a `non-resonant' set, which is filled up to an exponentially small set of measure $e^{-c K}$ (where $K$ is the maximal size of resonances considered) by primary maximal KAM tori; a `simply resonant set' of measure $\sqrt{\varepsilon} K^a$ and a third set of measure $\varepsilon K^b$ which is `non perturbative', in the sense that the $H$-dynamics on it can be described by a natural system which is {\sl not} nearly-integrable. We then focus on the simply resonant set -- the dynamics of which is particularly interesting (e.g., for Arnol'd diffusion, or the existence of secondary tori) -- and show that on such a set the secular (averaged) 1 degree-of-freedom Hamiltonians (labelled by the resonance index $k\in\mathbb{Z}^n$) can be put into a universal form (which we call `Generic Standard Form'), whose main analytic properties are controlled by {\sl only one parameter, which is uniform in the resonance label $k$}

On the measure of KAM tori in two degrees of freedom

A conjecture of Arnold, Kozlov and Neishtadt on the exponentially small measure of the non-torus set in analytic systems with two degrees of freedom is discussed

Prognostic role of atrial fibrillation in acute coronary syndromes: a real-life, contemporary analysis

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Prognostic value of low heart rates in patients admitted with acute myocardial infarction

INTRODUCTION AND OBJECTIVES: The risk prediction scores adopted in acute coronary syndromes (ACS) use incremental models to estimate mortality for heart rate (HR) above 60 bpm. Nonetheless, previous studies reported a nonlinear relationship between HR and events, suggesting that low HR may have an unrecognized prognostic role. We aimed to assess the prognostic impact of low HR in ACS, defined as admission HR <50 bpm. METHODS: This study analyzed data from the AMIS Plus registry, a cohort of hospitalized patients with ACS between 1999 and 2021. The primary endpoint was in-hospital all-cause mortality, while a composite of all-cause mortality, major cardiac/cerebrovascular events was set as the secondary endpoint. A multilevel statistical method was used to assess the prognostic role of low HR in ACS. RESULTS: The study included 51 001 patients. Crude estimates showed a bimodal distribution of primary and secondary endpoints with peaks at low and high HR. A nonlinear relationship between HR and in-hospital mortality was observed on restricted cubic spline analysis. An HR of 50 to 75 bpm showed lower mortality than HR <50 bpm (OR, 0.67; 95%CI, 0.47-0.99) only after primary multivariable analysis, which was not confirmed after multiple sensitivity analyses. After propensity score matching, progressive fading of the prognostic role of HR <50 bpm was evident. CONCLUSIONS: Low admission HR in ACS is associated with a higher crude rate of adverse events. Nonetheless, after correction for baseline differences, the prognostic role of low HR was not confirmed. Therefore, low HR probably represents a marker of underlying morbidity. These results may be clinically relevant in improving the accuracy of risk scores in ACS