Families of classical subgroup separable superintegrable systems

We describe a method for determining a complete set of integrals for a classical Hamiltonian that separates in orthogonal subgroup coordinates. As examples, we use it to determine complete sets of integrals, polynomial in the momenta, for some families of generalized oscillator and Kepler-Coulomb systems, hence demonstrating their superintegrability. The latter generalizes recent results of Verrier and Evans, and Rodriguez, Tempesta and Winternitz. Another example is given of a superintegrable system on a non-conformally flat space.Comment: 9 page

Second order superintegrable systems in conformally flat spaces. IV. The classical 3D Stäckel transform and 3D classification theory

This article is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems, both classical and quantum, in conformally flat spaces. In the first part of the article we study the Stäckel transform (or coupling constant metamorphosis) as an invertible mapping between classical superintegrable systems on different three-dimensional spaces. We show first that all superintegrable systems with nondegenerate potentials are multiseparable and then that each such system on any conformally flat space is Stäckel equivalent to a system on a constant curvature space. In the second part of the article we classify all the superintegrable systems that admit separation in generic coordinates. We find that there are eight families of these systems

Italian viticulture: A multi-faceted model of development and regression.

Wine can be considered a niche product on the drinks market, due to the annual turnover it generates and the dispersion of the productive matrix which controls its production. However, it takes on a symbolic value compared to other drinks and boasts a unique link with the territory. Typicity translated into territorial values has brought about the success of winemaking regions of great national and international fame. However, it is necessary to communicate the values and adopt the consequent measures for other areas in a secondary position or with an intermediate development. The value of wine ex-cellar, unbottled and before vat has been analyzed taking into account production in different market segments, a basic element for estimating the value of the GSP per hectare of vineyard. This paper is based on the study of physical, economic and motivational parameters that determine the primary value, the cause for consolidating or abandoning winegrowing. Decisions made by the vinegrower translate into nursery demand which, in turn, determines the varieties and surface areas of the future vineyard

Quantum models related to fouled Hamiltonians of the harmonic oscillator

We study a pair of canonoid (fouled) Hamiltonians of the harmonic oscillator which provide, at the classical level, the same equation of motion as the conventional Hamiltonian. These Hamiltonians, say $K_{1}$ and $K_{2}$, result to be explicitly time-dependent and can be expressed as a formal rotation of two cubic polynomial functions, $H_{1}$ and $H_{2}$, of the canonical variables (q,p). We investigate the role of these fouled Hamiltonians at the quantum level. Adopting a canonical quantization procedure, we construct some quantum models and analyze the related eigenvalue equations. One of these models is described by a Hamiltonian admitting infinite self-adjoint extensions, each of them has a discrete spectrum on the real line. A self-adjoint extension is fixed by choosing the spectral parameter $\epsilon$ of the associated eigenvalue equation equal to zero. The spectral problem is discussed in the context of three different representations. For $\epsilon =0$, the eigenvalue equation is exactly solved in all these representations, in which square-integrable solutions are explicity found. A set of constants of motion corresponding to these quantum models is also obtained. Furthermore, the algebraic structure underlying the quantum models is explored. This turns out to be a nonlinear (quadratic) algebra, which could be applied for the determination of approximate solutions to the eigenvalue equations.Comment: 24 pages, no figures, accepted for publication on JM

Superintegrability in a two-dimensional space of nonconstant curvature

A Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively studied in the case of two-dimensional spaces of constant (possibly zero) curvature when all the independent integrals are either quadratic or linear in the canonical momenta. In this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions. This is done by examining in detail one of the spaces of revolution found by G. Koenigs. We determine that there are essentially three distinct potentials which when added to the free Hamiltonian of this space have this type of superintegrability. Separation of variables for the associated Hamilton–Jacobi and Schrödinger equations is discussed. The classical and quantum quadratic algebras associated with each of these potentials are determined

Thermostatistics in the neighborhood of the π\pi-mode solution for the Fermi-Pasta-Ulam β\beta system: from weak to strong chaos

We consider a $\pi$-mode solution of the Fermi-Pasta-Ulam $\beta$ system. By perturbing it, we study the system as a function of the energy density from a regime where the solution is stable to a regime, where is unstable, first weakly and then strongly chaotic. We introduce, as indicator of stochasticity, the ratio $\rho$ (when is defined) between the second and the first moment of a given probability distribution. We will show numerically that the transition between weak and strong chaos can be interpreted as the symmetry breaking of a set of suitable dynamical variables. Moreover, we show that in the region of weak chaos there is numerical evidence that the thermostatistic is governed by the Tsallis distribution.Comment: 15 pages, 5 figure

Gender-related time course of sleep disturbances and psychological symptoms during the COVID-19 lockdown: a longitudinal study on the Italian population

Italy was the first western hotspot of the COVID-19 pandemic. In order to contain the spread of the virus, the Italian Government imposed home confinement to the entire population for almost two months. The present study is the first large-scale longitudinal report of the sleep and mental health changes during the prolonged lockdown due to the COVID-19 outbreak. We focused on the gendered vulnerability in a sample of the Italian population since cross-sectional research identified women to be more at-risk than men during this unprecedented situation. A total of 2701 individuals (mean age ± standard deviation, 32.37 ± 11.62; range, 18–82) participated in a web-based longitudinal survey consisting of two measurements. Participants were first-time recruited on social networks and via telephone messages through a snowball sampling and tested during the third week of the lockdown period. Subsequently, a follow-up evaluation was carried out during the seventh week of restraining measures. The survey assessed sleep quality, insomnia and depression symptoms, perceived stress, and anxiety, using the following questionnaires: the Pittsburgh Sleep Quality Index, the Insomnia Severity Index, the Beck Depression Inventory-second edition, the 10-item Perceived Stress Scale, and the State-Anxiety Inventory. Female gender showed the worst condition for all the examined dimensions in both the assessments. Nevertheless, at the follow-up women reported a reduction in insomnia and depression severity symptoms, perceived stress, and anxiety. On the other hand, male participants showed a worsening of sleep quality, insomnia symptoms, and perceived stress. Consequently, the gender prevalence gap of clinical conditions such as insomnia and depression was largely reduced under lockdown. Our investigation pointed to a different time course of sleep and mental health between genders during the home confinement period. Women seemed to show greater long-term resilience during the lockdown. Meanwhile, the male gender emerges as the most vulnerable category to the extension of the restraining measures. Our results suggest that there is no “weaker gender” after a prolonged lockdown. Indeed, the Italian population transversely presented signs of psychological suffering and significant sleep disturbances after the protracted and stressful lockdown period due to the COVID-19 pandemic