Realization of the Three-dimensional Quantum Euclidean Space by Differential Operators

The three-dimensional quantum Euclidean space is an example of a non-commutative space that is obtained from Euclidean space by $q$-deformation. Simultaneously, angular momentum is deformed to $so_q(3)$, it acts on the $q$-Euclidean space that becomes a $so_q(3)$-module algebra this way. In this paper it is shown, that this algebra can be realized by differential operators acting on $C^{\infty}$ functions on $\mathbb{R}^3$. On a factorspace of $C^{\infty}(\mathbb{R}^3)$ a scalar product can be defined that leads to a Hilbert space, such that the action of the differential operators is defined on a dense set in this Hilbert space and algebraically self-adjoint becomes self-adjoint for the linear operator in the Hilbert space. The self-adjoint coordinates have discrete eigenvalues, the spectrum can be considered as a $q$-lattice.Comment: 13 pages, late

The impact of the social environment on children's mental health in a prosperous city: an analysis with data from the city of Munich

BACKGROUND: Children with a low socioeconomic position are more affected by mental difficulties as compared to children with a higher socioeconomic position. This paper explores whether this socioeconomic pattern persists in the prosperous German city of Munich which features high quality of life and coverage of children mental health specialists that lies well above the national average and is among the highest in Europe. METHODS: 1,265 parents of preschool children participated in a cross-sectional health survey. They were given a self-administered questionnaire (including socioeconomic variables) and the 'Strengths and Difficulties Questionnaire (SDQ)', a well-established method to identify mental difficulties among children and adolescents. Prevalence estimates for the 'SDQ-Total Difficulties Score' were calculated, with a special focus on differences by parental (resp. household) socioeconomic position. The association between parental education, household income, single parenthood, nationality, and parental working status on one hand, and their children's mental health on the other, was explored using multivariable logistic regression models. The coverage of mental health specialists per 100,000 children aged 14 or younger in the city of Munich was also calculated. RESULTS: In Munich, the distribution of mental health difficulties among children follows the same socioeconomic pattern as described previously at the national level, but the overall prevalence is about 30% lower. Comparing different indicators of socioeconomic position, low parental education and household income are the strongest independent variables associated with mental difficulties among children (OR = 2.7; CI = 1.6 - 4.4 and OR = 2.8; CI = 1.4 - 5.6, respectively). CONCLUSIONS: Socioeconomic differences in the prevalence of childhood mental difficulties are very stable. Even in a city such as Munich, which is characterized by high quality of life, high availability of mental health specialists, and low overall prevalence of these mental difficulties, they are about as pronounced as in Germany as a whole. It can be concluded that the effect of several characteristics of socioeconomic position 'overrules' the effect of a health promoting regional environment

ω(→π+π−π0) \omega (\to \pi^+\pi^-\pi^0) meson photoproduction on proton

The cross section is estimated for the $\pi^+\pi^-\pi^0$ invariant mass distribution in the $\gamma p$ reaction in the GeV region. This reaction is assumed to proceed through the formation of the $\omega$ meson in the intermediate state, since the production cross section for this meson in the $\gamma p$ reaction in GeV region is significant and it has large branching ratio (88.8%) in the $\pi^+\pi^-\pi^0$ channel. The cross sections for this reaction have been calculated using the energy dependent reaction amplitude, i.e., $f_{\gamma p \to \omega p}(0)$, extracted from the latest $\omega$ meson photoproduction data. We use established procedure to calculate other factors, like width and propagator of the $\omega$ meson, so that our calculation can provide reliable cross section. The calculated results reproduce the measured $\pi^+\pi^-\pi^0$ invariant mass distribution spectra in the $\gamma p$ reaction.Comment: 13 pages, 5 figure