Spacetime Encodings II - Pictures of Integrability

I visually explore the features of geodesic orbits in arbitrary stationary axisymmetric vacuum (SAV) spacetimes that are constructed from a complex Ernst potential. Some of the geometric features of integrable and chaotic orbits are highlighted. The geodesic problem for these SAV spacetimes is rewritten as a two degree of freedom problem and the connection between current ideas in dynamical systems and the study of two manifolds sought. The relationship between the Hamilton-Jacobi equations, canonical transformations, constants of motion and Killing tensors are commented on. Wherever possible I illustrate the concepts by means of examples from general relativity. This investigation is designed to build the readers' intuition about how integrability arises, and to summarize some of the known facts about two degree of freedom systems. Evidence is given, in the form of orbit-crossing structure, that geodesics in SAV spacetimes might admit, a fourth constant of motion that is quartic in momentum (by contrast with Kerr spacetime, where Carter's fourth constant is quadratic).Comment: 11 pages, 10 figure

Ueber Condensationen von Chloral und Butychloral mit Aceton und Acetophenon

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Nondegenerate 3D complex Euclidean superintegrable systems and algebraic varieties

A classical (or quantum) second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n-1 functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multi-separability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schroedinger operator, deep connections with special functions and with QES systems. Here we announce a complete classification of nondegenerate (i.e., 4-parameter) potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in 10 variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly 10 nondegenerate potentials.Comment: 35 page

Damage to the prefrontal cortex increases utilitarian moral judgements

The psychological and neurobiological processes underlying moral judgement have been the focus of many recent empirical studies1–11. Of central interest is whether emotions play a causal role in moral judgement, and, in parallel, how emotion-related areas of the brain contribute to moral judgement. Here we show that six patients with focal bilateral damage to the ventromedial prefrontal cortex (VMPC), a brain region necessary for the normal generation of emotions and, in particular, social emotions12–14, produce an abnor- mally ‘utilitarian’ pattern of judgements on moral dilemmas that pit compelling considerations of aggregate welfare against highly emotionally aversive behaviours (for example, having to sacrifice one person’s life to save a number of other lives)7,8. In contrast, the VMPC patients’ judgements were normal in other classes of moral dilemmas. These findings indicate that, for a selective set of moral dilemmas, the VMPC is critical for normal judgements of right and wrong. The findings support a necessary role for emotion in the generation of those judgements

New superintegrable models with position-dependent mass from Bertrand's Theorem on curved spaces

A generalized version of Bertrand's theorem on spherically symmetric curved spaces is presented. This result is based on the classification of (3+1)-dimensional (Lorentzian) Bertrand spacetimes, that gives rise to two families of Hamiltonian systems defined on certain 3-dimensional (Riemannian) spaces. These two systems are shown to be either the Kepler or the oscillator potentials on the corresponding Bertrand spaces, and both of them are maximally superintegrable. Afterwards, the relationship between such Bertrand Hamiltonians and position-dependent mass systems is explicitly established. These results are illustrated through the example of a superintegrable (nonlinear) oscillator on a Bertrand-Darboux space, whose quantization and physical features are also briefly addressed.Comment: 13 pages; based in the contribution to the 28th International Colloquium on Group Theoretical Methods in Physics, Northumbria University (U.K.), 26-30th July 201

Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: II. Darboux Spaces DIII and DIV

This is the second paper on the path integral approach of superintegrable systems on Darboux spaces, spaces of non-constant curvature. We analyze in the spaces \DIII and \DIV five respectively four superintegrable potentials, which were first given by Kalnins et al. We are able to evaluate the path integral in most of the separating coordinate systems, leading to expressions for the Green functions, the discrete and continuous wave-functions, and the discrete energy-spectra. In some cases, however, the discrete spectrum cannot be stated explicitly, because it is determined by a higher order polynomial equation. We show that also the free motion in Darboux space of type III can contain bound states, provided the boundary conditions are appropriate. We state the energy spectrum and the wave-functions, respectively

Mathematically Gifted Adolescents Have Deficiencies in Social Valuation and Mentalization

Many mathematically gifted adolescents are characterized as being indolent, underachieving and unsuccessful despite their high cognitive ability. This is often due to difficulties with social and emotional development. However, research on social and emotional interactions in gifted adolescents has been limited. The purpose of this study was to observe differences in complex social strategic behaviors between gifted and average adolescents of the same age using the repeated Ultimatum Game. Twenty-two gifted adolescents and 24 average adolescents participated in the Ultimatum Game. Two adolescents participate in the game, one as a proposer and the other as a responder. Because of its simplicity, the Ultimatum Game is an apt tool for investigating complex human emotional and cognitive decision-making in an empirical setting. We observed strategic but socially impaired offers from gifted proposers and lower acceptance rates from gifted responders, resulting in lower total earnings in the Ultimatum Game. Thus, our results indicate that mathematically gifted adolescents have deficiencies in social valuation and mentalization

A ∼\sim15 kpc outflow cone piercing through the halo of the blue compact metal-poor galaxy SBS0335-052

Context: Outflows from low-mass star-forming galaxies are a fundamental ingredient for models of galaxy evolution and cosmology. Aims: The onset of kpc-scale ionised filaments in the halo of the metal-poor compact dwarf SBS 0335-052E was previously not linked to an outflow. We here we investigate whether these filaments provide evidence for an outflow. Methods: We obtained new VLT/MUSE WFM and deep NRAO/VLA B-configuration 21cm data of the galaxy. The MUSE data provide morphology, kinematics, and emission line ratios H$\beta$/H$\alpha$ and [\ion{O}{iii}]$\lambda5007$/H$\alpha$ of the low surface-brightness filaments, while the VLA data deliver morphology and kinematics of the neutral gas in and around the system. Both datasets are used in concert for comparisons between the ionised and the neutral phase. Results: We report the prolongation of a lacy filamentary ionised structure up to a projected distance of 16 kpc at $\mathrm{SB}_\mathrm{H\alpha} = 1.5\times10^{-18}$erg s$^{-1}$ cm$^{-2}$arcsec$^{-2}$. The filaments exhibit unusual low H$\alpha$/H$\beta \approx 2.4$ and low [\ion{O}{iii}]/H$\alpha \sim 0.4 - 0.6$ typical of diffuse ionised gas. They are spectrally narrow ($\sim 20$ km s$^{-1}$) and exhibit no velocity sub-structure. The filaments extend outwards of the elongated \ion{H}{I} halo. On small scales the $N_\mathrm{HI}$ peak is offset from the main star-forming sites. Morphology and kinematics of \ion{H}{I} and \ion{H}{II} reveal how star-formation driven feedback interacts differently with the ionised and the neutral phase. Conclusions: We reason that the filaments are a large scale manifestation of star-formation driven feedback, namely limb-brightened edges of a giant outflow cone that protrudes through the halo of this gas-rich system. A simple toy model of such a conical-structure is found to be commensurable with the observations.Comment: Accepted version in A&A after language editing. 22 pages, 24 figure