Algebraic nonlinear collective motion

Finite-dimensional Lie algebras of vector fields determine geometrical collective models in quantum and classical physics. Every set of vector fields on Euclidean space that generates the Lie algebra sl(3, R) and contains the angular momentum algebra so(3) is determined. The subset of divergence-free sl(3, R) vector fields is proven to be indexed by a real number $\Lambda$. The $\Lambda=0$ solution is the linear representation that corresponds to the Riemann ellipsoidal model. The nonlinear group action on Euclidean space transforms a certain family of deformed droplets among themselves. For positive $\Lambda$, the droplets have a neck that becomes more pronounced as $\Lambda$ increases; for negative $\Lambda$, the droplets contain a spherical bubble of radius $|\Lambda|^{{1/3}}$. The nonlinear vector field algebra is extended to the nonlinear general collective motion algebra gcm(3) which includes the inertia tensor. The quantum algebraic models of nonlinear nuclear collective motion are given by irreducible unitary representations of the nonlinear gcm(3) Lie algebra. These representations model fissioning isotopes ($\Lambda>0$) and bubble and two-fluid nuclei ($\Lambda<0$).Comment: 32pages, 4 figures not include

Galactic oscillator symmetry

Riemann ellipsoids model rotating galaxies when the galactic velocity field is a linear function of the Cartesian coordinates of the galactic masses. In nuclear physics, the kinetic energy in the linear velocity field approximation is known as the collective kinetic energy. But, the linear approximation neglects intrinsic degrees of freedom associated with nonlinear velocity fields. To remove this limitation, the theory of symplectic dynamical symmetry is developed for classical systems. A classical phase space for a self-gravitating symplectic system is a co-adjoint orbit of the noncompact group SP(3,R). The degenerate co-adjoint orbit is the 12 dimensional homogeneous space Sp(3,R)/U(3), where the maximal compact subgroup U(3) is the symmetry group of the harmonic oscillator. The Hamiltonian equations of motion on each orbit form a Lax system X = (X,F), where X and F are elements of the symplectic Lie algebra. The elements of the matrix X are the generators of the symplectic Lie algebra, viz., the one-body collective quadratic functions of the positions and momenta of the galactic masses. The matrix F is composed from the self-gravitating potential energy, the angular velocity, and the hydostatic pressure. Solutions to the hamiltonian dynamical system on Sp(3,R)/U(3) are given by symplectic isospectral deformations. The Casimirs of Sp(3,R), equal to the traces of powers of X, are conserved quantities

Self-consistent anisotropic oscillator with cranked angular and vortex velocities

The Kelvin circulation is the kinematical Hermitian observable that measures the true character of nuclear rotation. For the anisotropic oscillator, mean field solutions with fixed angular momentum and Kelvin circulation are derived in analytic form. The cranking Lagrange multipliers corresponding to the two constraints are the angular and vortex velocities. Self-consistent solutions are reported with a constraint to constant volume.Comment: 12 pages, LaTex/RevTex, Phys. Rev. C4

Riemann's theorem for quantum tilted rotors

The angular momentum, angular velocity, Kelvin circulation, and vortex velocity vectors of a quantum Riemann rotor are proven to be either (1) aligned with a principal axis or (2) lie in a principal plane of the inertia ellipsoid. In the second case, the ratios of the components of the Kelvin circulation to the corresponding components of the angular momentum, and the ratios of the components of the angular velocity to those of the vortex velocity are analytic functions of the axes lengths.Comment: 8 pages, Phys. Rev.

Nonlinear collective nuclear motion

For each real number $\Lambda$ a Lie algebra of nonlinear vector fields on three dimensional Euclidean space is reported. Although each algebra is mathematically isomorphic to $gl(3,{\bf R})$, only the $\Lambda=0$ vector fields correspond to the usual generators of the general linear group. The $\Lambda < 0$ vector fields integrate to a nonstandard action of the general linear group; the $\Lambda >0$ case integrates to a local Lie semigroup. For each $\Lambda$, a family of surfaces is identified that is invariant with respect to the group or semigroup action. For positive $\Lambda$ the surfaces describe fissioning nuclei with a neck, while negative $\Lambda$ surfaces correspond to exotic bubble nuclei. Collective models for neck and bubble nuclei are given by irreducible unitary representations of a fifteen dimensional semidirect sum spectrum generating algebra $gcm(3)$ spanned by its nonlinear $gl(3,{\bf R})$ subalgebra plus an abelian nonlinear inertia tensor subalgebra.Comment: 13 pages plus two figures(available by fax from authors by request

A Predictive and Causal-Comparative Analysis of Teacher Burnout and Emotional Empathy among K-12 Public School Teachers

Recent research has shown that teacher burnout is a contributing factor to many teachers leaving the field of education early on in their careers. Many teachers of all levels and subjects leave before ever reaching their full potential due to decreased job-satisfaction and the overwhelming symptoms of burnout crippling other areas of their life. Emotional exhaustion is a major component of burnout. Thus, emotional empathy may be related to teacher burnout. In this quantitative, correlational and causal-comparative study, the researcher examined the relationship between emotional empathy and teacher burnout among K-12 teachers. Further, the researcher investigated whether the relationship between emotional empathy and burnout was more significant among teachers of different instructional assignments. The researcher drew from a sample of 50 regular education and 50 special education teachers from five, rural school districts. Teachers completed a set of instruments: the Maslach Burnout MBI-Educators Survey (MBI-ES) and The Questionnaire Measure of Emotional Empathy (QMEE), via an online survey. Using a linear regression, the researcher examined the relationship between the participant scores on the Questionnaire Measure of Emotional Empathy and the participant scores on Maslach Burnout Inventory-Educator’s Survey scores. Using an independent samples t-test, the researcher also compared the levels of burnout and empathy among regular education teachers and special education teachers. A significant relationship was found between emotional empathy and burnout among both regular education and special education teachers. There was no significant difference between emotional empathy or burnout scores of regular and special education teachers

Some physical and thermodynamic properties of rocket exhaust clouds measured with infrared scanners

Measurements using infrared scanners were made of the radiation from exhaust clouds from liquid- and solid-propellant rocket boosters. Field measurements from four launches were discussed. These measurements were intended to explore the physical and thermodynamic properties of these exhaust clouds during their formation and subsequent dispersion. Information was obtained concerning the initial cloud's buoyancy, the stabilized cloud's shape and trajectory, the cloud volume as a function of time, and it's initial and stabilized temperatures. Differences in radiation intensities at various wavelengths from ambient and stabilized exhaust clouds were investigated as a method of distinguishing between the two types of clouds. The infrared remote sensing method used can be used at night when visible range cameras are inadequate. Infrared scanning techniques developed in this project can be applied directly to natural clouds, clouds containing certain radionuclides, or clouds of industrial pollution