Quantization for uniform distributions on equilateral triangles

We approximate the uniform measure on an equilateral triangle by a measure supported on $n$ points. We find the optimal sets of points ($n$-means) and corresponding approximation (quantization) error for $n\leq4$, give numerical optimization results for $n\leq 21$, and a bound on the quantization error for $n\to\infty$. The equilateral triangle has particularly efficient quantizations due to its connection with the triangular lattice. Our methods can be applied to the uniform distributions on general sets with piecewise smooth boundaries

Energy levels, radiative rates and electron impact excitation rates for transitions in He-like Ga XXX, Ge XXXI, As XXXII, Se XXXIII and Br XXXIV

We report calculations of energy levels, radiative rates and electron impact excitation cross sections and rates for transitions in He-like Ga XXX, Ge XXXI, As XXXII, Se XXXIII and Br XXXIV. The {\sc grasp} (general-purpose relativistic atomic structure package) is adopted for calculating energy levels and radiative rates. For determining the collision strengths, and subsequently the excitation rates, the Dirac Atomic R-matrix Code ({\sc darc}) is used. Oscillator strengths, radiative rates and line strengths are reported for all E1, E2, M1 and M2 transitions among the lowest 49 levels of each ion. Additionally, theoretical lifetimes are provided for all 49 levels of the above five ions. Collision strengths are averaged over a Maxwellian velocity distribution and the effective collision strengths obtained listed over a wide temperature range up to 10$^{8}$ K. Comparisons are made with similar data obtained using the Flexible Atomic Code ({\sc fac}) to highlight the importance of resonances, included in calculations with {\sc darc}, in the determination of effective collision strengths. Discrepancies between the collision strengths from {\sc darc} and {\sc fac}, particularly for some forbidden transitions, are also discussed. Finally, discrepancies between the present results for effective collision strengths with the {\sc darc} code and earlier semi-relativistic $R$-matrix data are noted over a wide range of electron temperatures for many transitions in all ions.Comment: 11 pages of Text, 11 Figures and 4 Tables. Ref: Physica Scripta 87 (2013) in press. arXiv admin note: substantial text overlap with arXiv:1207.6525, arXiv:1209.2914, arXiv:1207.542

Black-Hole Solutions with Scalar Hair in Einstein-Scalar-Gauss-Bonnet Theories

In the context of the Einstein-scalar-Gauss-Bonnet theory, with a general coupling function between the scalar field and the quadratic Gauss-Bonnet term, we investigate the existence of regular black-hole solutions with scalar hair. Based on a previous theoretical analysis, that studied the evasion of the old and novel no-hair theorems, we consider a variety of forms for the coupling function (exponential, even and odd polynomial, inverse polynomial, and logarithmic) that, in conjunction with the profile of the scalar field, satisfy a basic constraint. Our numerical analysis then always leads to families of regular, asymptotically-flat black-hole solutions with non-trivial scalar hair. The solution for the scalar field and the profile of the corresponding energy-momentum tensor, depending on the value of the coupling constant, may exhibit a non-monotonic behaviour, an unusual feature that highlights the limitations of the existing no-hair theorems. We also determine and study in detail the scalar charge, horizon area and entropy of our solutions.Comment: PdfLatex file, 29 Pages, 18 figures, the analysis was extended to study the scalar charge, horizon area and entropy of our solutions, comments added, typos corrected, version to appear in Physical Review

Hawking Radiation from a (4+n)-Dimensional Rotating Black Hole on the Brane

We study the emission of Hawking radiation in the form of scalar fields from a (4+n)-dimensional, rotating black hole on the brane. We perform a numerical analysis to solve both the radial and angular parts of the scalar field equation, and derive exact results for the radial wavefunction and angular eigenvalues, respectively. We then determine the Hawking radiation energy emission rate, and find that, as the angular momentum increases, it is suppressed in the low-energy regime but enhanced in the intermediate and high-energy regimes. Our results agree with previous analytical studies, derived in the low-angular momentum and low-energy approximation, and generalize them to include angular momentum and energy regimes that were until now unexplored. We also investigate the energy amplification due to super-radiance and we find that, in the presence of extra dimensions, the effect is significantly enhanced.Comment: 9 pages, Latex file, 5 figures, a new figure and a paragraph have been added along with some clarifying comments, version to appear in Phys. Lett.