486 research outputs found
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 -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,
and -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
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