W-Symmetries of Ito stochastic differential equations

We discuss W-symmetries of Ito stochastic differential equations, introduced in a recent paper by Gaeta and Spadaro [J. Math. Phys. 2017]. In particular, we discuss the general form of acceptable generators for continuous (Lie-point) W-symmetry, arguing they are related to the (linear) conformal group, and how W-symmetries can be used in the integration of Ito stochastic equations along Kozlov theory for standard (deterministic or random) symmetries. It turns out this requires, in general, to consider more general classes of stochastic equations than just Ito ones.Comment: Preprint version; final (improved) version to appear in J. Math. Phy

Twisted symmetries of differential equations

We review the basic ideas lying at the foundation of the recently developed theory of twisted symmetries of differential equations, and some of its developments

A simple SIR model with a large set of asymptomatic infectives

There is increasing evidence that one of the most difficult problems in trying to control the ongoing COVID-19 epidemic is the presence of a large cohort of asymptomatic infectives. We develop a SIR-type model taking into account the presence of asymptomatic, or however undetected, infective, and the substantially long time these spend being infective and not isolated. We discuss how a SIR-based prediction of the epidemic course based on early data but not taking into account the presence of a large set of asymptomatic infectives would give wrong estimate of very relevant quantities such as the need of hospital beds, the time to the epidemic peak, and the number of people which are left untouched by the first wave and thus in danger in case of a second epidemic wave. In the second part of the note, we apply our model to the COVID-19 epidemics in Italy. We obtain a good agreement with epidemiological data; according to the best fit of epidemiological data in terms of this model, only 10\% of infectives in Italy is symptomatic.Comment: V4 (hopefully final) contains analysis of data up to May 15, 202

Symmetry of stochastic equations

Symmetry methods are by now recognized as one of the main tools to attack deterministic differential equations (both ODEs and PDEs); the situation is quite different for what concerns stochastic differential equations: here, symmetry considerations are of course quite widely used by theoretical physicists, but a rigorous and general theory comparable to the one developed for deterministic equation is still lacking. In the following I will report on some work I have done on symmetries of stochastic (Ito) equations, and how these compare with the symmetries of the associated diffusion (Fokker-Planck) equations.Comment: Work prepared for the Kyev SNMP2003 conference proceeding

Simple and collective twisted symmetries

After the introduction of $\lambda$-symmetries by Muriel and Romero, several other types of so called "twisted symmetries" have been considered in the literature (their name refers to the fact they are defined through a deformation of the familiar prolongation operation); they are as useful as standard symmetries for what concerns symmetry reduction of ODEs or determination of special (invariant) solutions for PDEs and have thus attracted attention. The geometrical relation of twisted symmetries to standard ones has already been noted: for some type of twisted symmetries (in particular, $\lambda$ and $\mu$-symmetries), this amounts to a certain kind of gauge transformation. In a previous review paper [G. Gaeta, "Twisted symmetries of differential equations", {\it J. Nonlin. Math. Phys.}, {\bf 16-S} (2009), 107-136] we have surveyed the first part of the developments of this theory; in the present paper we review recent developments. In particular, we provide a unifying geometrical description of the different types of twisted symmetries; this is based on the classical Frobenius reduction applied to distribution generated by Lie-point (local) symmetries.Comment: 40 pages; to appear in J. Nonlin. Math. Phys. 21 (2014), 593-62

The Poincare'-Nekhoroshev map

We study a generalization of the familiar Poincar\'e map, first implicitely introduced by N.N. Nekhoroshev in his study of persistence of invariant tori in hamiltonian systems, and discuss some of its properties and applications. In particular, we apply it to study persistence and bifurcation of invariant tori.Comment: arxiv version is already officia