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

High resolution single cell profiling of human hematopoietic stem cell drug products

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Periodic orbits close to elliptic tori and applications to the three-body problem

We prove, under suitable non-resonance and non-degeneracy ``twist'' conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori (of Hamiltonian systems). We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricity-inclination regime. Furthermore, we find other periodic orbits under some restrictions on the period and the masses of the ``planets''. The proofs are based on averaging theory, KAM theory and variational methods. (Supported by M.U.R.S.T. Variational Methods and Nonlinear Differential Equations.

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

Retroviral Integrations in Gene Therapy Trials

γ-Retroviral and lentiviral vectors allow the permanent integration of a therapeutic transgene in target cells and have provided in the last decade a delivery platform for several successful gene therapy (GT) clinical approaches. However, the occurrence of adverse events due to insertional mutagenesis in GT treated patients poses a strong challenge to the scientific community to identify the mechanisms at the basis of vector-driven genotoxicity. Along the last decade, the study of retroviral integration sites became a fundamental tool to monitor vector–host interaction in patients overtime. This review is aimed at critically revising the data derived from insertional profiling, with a particular focus on the evidences collected from GT clinical trials. We discuss the controversies and open issues associated to the interpretation of integration site analysis during patient's follow up, with an update on the latest results derived from the use of high-throughput technologies. Finally, we provide a perspective on the future technical development and on the application of these studies to address broader biological questions, from basic virology to human hematopoiesis

KAM theory for the Hamiltonian derivative wave equation

We prove an infinite dimensional KAM theorem which implies the existence of Cantor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equationsComment: 66 page