arXiv subjects
Publications and source records attributed to Giuseppe Gaeta.
The notion of gauge transform has its origin in Physics (Field Theory). In the present note we discuss -- from a purely mathematical perspective -- special gauge transforms of autonomous first order ODE's and their special properties. Particular attention is given to the problem of identifying those nonautonomous ODE's which are gauge transforms of autonomous systems.
The classical Floquet theory allows to map a time-periodic system of linear differential equations into an autonomous one. By looking at it in a geometrical way, we extend the theory to a class of non-autonomous non-periodic equations. This is obtained by considering a change of variables which depends on time in a non-trivial way, i.e. introducing gauge transformations, well known in fundamental Physics and Field Theory -- but which seems to have received little attention in this context.
Conditional symmetries were introduced by Levi and Winternitz in their 1989 seminal paper to deal with nonlinear PDEs. Here we discuss their application in the framework of ODEs, and more specifically Dynamical Systems; it turns out they are closely related to two established -- albeit maybe less widely known -- concepts, i.e. orbital symmetries and configurational invariants. The paper is devoted to studying the interplay of these notions, and their application in the study of Dynamical Systems, with special attention to invariant manifolds of these.
We present a version of the classical Moran model, in which mutations are taken into account; the possibility of mutations was introduced by Moran in his seminal paper, but it is more often overlooked in discussing the Moran model. For this model, fixation is prevented by mutation, and we have an ergodic Markov process; the equilibrium distribution for such a process was determined by Moran. The problems we consider in this paper are those of first hitting either one of the ``pure'' (uniform population) states, depending on the initial state; and that of first hitting times. The presence of mutations leads to a nonlinear dependence of the hitting probabilities on the initial state, and to a larger mean hitting time compared to the mutation-free process (in which case hitting corresponds to fixation of one of the alleles).
Symmetries can be used to integrate scalar Ito equation -- or reduce systems of such equations -- by the Kozlov substitution, i.e. passing to symmetry adapted coordinates. While the theory is well established for so called deterministic standard symmetries (the class originally studied by Kozlov), some points need clarification for so called random standard symmetries and W-symmetries. This paper is devoted to such clarification; in particular we note that the theory naturally calls, for these classes of symmetries, to also consider generalized Ito equations; and that while Kozlov theory is extended substantially unharmed for random standard symmetries, W-symmetries should be handled with great care, and cannot be used towards integration of stochastic equations, albeit they have different uses.
In its most restrictive definition, an octupolar tensor is a fully symmetric traceless third-rank tensor in three space dimensions. So great a body of works have been devoted to this specific class of tensors and their physical applications that a review would perhaps be welcome by a number of students. Here, we endeavour to place octupolar tensors into a broader perspective, considering non-vanishing traces and non-fully symmetric tensors as well. A number of general concepts are recalled and applied to either octupolar and higher-rank tensors. As a tool to navigate the diversity of scenarios we envision, we introduce the octupolar potential, a scalar-valued function which can easily be given an instructive geometrical representation. Physical applications are plenty; those to liquid crystal science play a major role here, as they were the original motivation for our interest in the topic of this review.
In a recent paper we have classified scalar Ito equations which admits a standard symmetry; these are also directly integrable by the Kozlov substitution. In the present work, we consider the diffusion (Fokker-Planck) equations associated to such symmetric Ito equations.
The classification of scalar Ito equations with a single noise source which admit a so called standard symmetry and hence are -- by the Kozlov construction -- integrable is by now complete. In this paper we study the situation, occurring in physical as well as biological applications, where there are two independent noise sources. We determine all such autonomous Ito equations admitting a standard symmetry; we then integrate them by means of the Kozlov construction. We also consider the case of three or more independent noises, showing no standard symmetry is present.
It is known that knowledge of a symmetry of a scalar Ito stochastic differential equations leads, thanks to the Kozlov substitution, to its integration. In the present paper we provide a classification of scalar autonomous Ito stochastic differential equations with simple noise possessing symmetries; here "simple noise" means the noise coefficient is of the form $\s (x,t) = s x^k$, with $s$ and $k$ real constants. Such equations can be taken to a standard form via a well known transformation; for such standard forms we also provide the integration of the symmetric equations. Our work extends previous classifications in that it also consider recently introduced types of symmetries, in particular standard random symmetries, not considered in those.
Scientific research changed profoundly over the last 30 years, in all its aspects. Scientific publishing has changed as well, mainly because of the strong increased number of submitted papers and because of the appearance of Open Access journals and publishers. We propose some reflections on these issues.
Mass vaccination produces a reduction in virus circulation, but also evolutive pressure towards the appearance of virus-resistant strains. We discuss the balance between these two effects, in particular when the mass vaccination takes place in the middle of an epidemic period.
We classify simple symmetries for an Ornstein-Uhlenbeck process, describing a particle in an external force field $f(x)$. It turns out that for sufficiently regular (in a sense to be defined) forces there are nontrivial symmetries only if $f(x)$ is at most linear. We fully discuss the isotropic case, while for the non-isotropic we only deal with a generic situation (defined in detail in the text).
We consider several aspects of conjugating symmetry methods, including the method of invariants, with an asymptotic approach. In particular we consider how to extend to the stochastic setting several ideas which are well established in the deterministic one, such as conditional, partial and asymptotic symmetries. A number of explicit examples are presented.
Normal modes are intimately related to the quadratic approximation of a potential at its hyperbolic equilibria. Here we extend the notion to the case where the Taylor expansion for the potential at a critical point starts with higher order terms, and show that such an extension shares some of the properties of standard normal modes. Some symmetric examples are considered in detail.
Different countries -- and sometimes different regions within the same countries -- have adopted different strategies in trying to contain the ongoing COVID-19 epidemic; these mix in variable parts social confinement, early detection and contact tracing. In this paper we discuss the different effects of these ingredients on the epidemic dynamics; the discussion is conducted with the help of two simple models, i.e. the classical SIR model and the recently introduced variant (A-SIR, arXiv:2003.08720) which takes into account the presence of a large set of asymptomatic infectives.
The most important features to assess the severity of an epidemic are its size and its timescale. We discuss these features in a systematic way in the context of SIR and SIR-type models. We investigate in detail how the size and timescale of the epidemic can be changed by acting on the parameters characterizing the model. Using these results and having as guideline the COVID-19 epidemic in Italy, we compare the efficiency of different containment strategies for contrasting an epidemic diffusion such as social distancing, lockdown, tracing, early detection and isolation.
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.
Social distancing, often in the form of lockdown, has been adopted by many countries as a way to contrast the spreading of COVID-19. We discuss the temporal aspects of social distancing in contrasting an epidemic diffusion. We argue that a strategy based uniquely on social distancing requires to maintain the relative measures for a very long time, while a more articulate strategy, which also uses early detection and prompt isolation, can be both more efficient on reducing the epidemic peak and allow to relax the social distancing measures after a much shorter time. We consider in more detail the situation in Italy, simulating the effect of different strategies through a recently introduced SIR-type epidemiological model. The short answer to the question in the title is: "it depends on what else you do".