Distribution of certain minor elements in Alaskan coals

Seventy-five samples of coal from Northern Alaska, Jarvis Creek, Nenana, Matanuska, Kenai and Bering River Coal Fields were analyzed by quantitative spectrochemical procedures f o r lead, gallium, copper, barium, beryllium, nickel, titanium, vanadium, zirconium, cobalt, chromium, germanium, and tin. Other elements, of significance, identified from the spectrograms were, gold and silver identified in certain Nenana coals and silver in coals from Chickaloon in the Matanuska field, in concentrations up to several parts per million of coal ash. Forty-one of the above samples were sink-floated to study the distribution of minor elements between the organic and inorganic phases of the coals. Relative affinities of the minor of the minor elements to the organic matter in the coal is discussed

Some generic properties of level spacing distributions of 2D real random matrices

We study the level spacing distribution $P(S)$ of 2D real random matrices both symmetric as well as general, non-symmetric. In the general case we restrict ourselves to Gaussian distributed matrix elements, but different widths of the various matrix elements are admitted. The following results are obtained: An explicit exact formula for $P(S)$ is derived and its behaviour close to S=0 is studied analytically, showing that there is linear level repulsion, unless there are additional constraints for the probability distribution of the matrix elements. The constraint of having only positive or only negative but otherwise arbitrary non-diagonal elements leads to quadratic level repulsion with logarithmic corrections. These findings detail and extend our previous results already published in a preceding paper. For the {\em symmetric} real 2D matrices also other, non-Gaussian statistical distributions are considered. In this case we show for arbitrary statistical distribution of the diagonal and non-diagonal elements that the level repulsion exponent $\rho$ is always $\rho = 1$, provided the distribution function of the matrix elements is regular at zero value. If the distribution function of the matrix elements is a singular (but still integrable) power law near zero value of $S$, the level spacing distribution $P(S)$ is a fractional exponent pawer law at small $S$. The tail of $P(S)$ depends on further details of the matrix element statistics. We explicitly work out four cases: the constant (box) distribution, the Cauchy-Lorentz distribution, the exponential distribution and, as an example for a singular distribution, the power law distribution for $P(S)$ near zero value times an exponential tail.Comment: 21 pages, no figures, submitted to Zeitschrift fuer Naturforschung

Distribution of Scattering Matrix Elements in Quantum Chaotic Scattering

Scattering is an important phenomenon which is observed in systems ranging from the micro- to macroscale. In the context of nuclear reaction theory the Heidelberg approach was proposed and later demonstrated to be applicable to many chaotic scattering systems. To model the universal properties, stochasticity is introduced to the scattering matrix on the level of the Hamiltonian by using random matrices. A long-standing problem was the computation of the distribution of the off-diagonal scattering-matrix elements. We report here an exact solution to this problem and present analytical results for systems with preserved and with violated time-reversal invariance. Our derivation is based on a new variant of the supersymmetry method. We also validate our results with scattering data obtained from experiments with microwave billiards.Comment: Published versio

Emergence of skew distributions in controlled growth processes

Starting from a master equation, we derive the evolution equation for the size distribution of elements in an evolving system, where each element can grow, divide into two, and produce new elements. We then probe general solutions of the evolution quation, to obtain such skew distributions as power-law, log-normal, and Weibull distributions, depending on the growth or division and production. Specifically, repeated production of elements of uniform size leads to power-law distributions, whereas production of elements with the size distributed according to the current distribution as well as no production of new elements results in log-normal distributions. Finally, division into two, or binary fission, bears Weibull distributions. Numerical simulations are also carried out, confirming the validity of the obtained solutions.Comment: 9 pages, 3 figure

On the Anti-Wishart distribution

We provide the probability distribution function of matrix elements each of which is the inner product of two vectors. The vectors we are considering here are independently distributed but not necessarily Gaussian variables. When the number of components M of each vector is greater than the number of vectors N, one has a $N\times N$ symmetric matrix. When $M\ge N$ and the components of each vector are independent Gaussian variables, the distribution function of the $N(N+1)/2$ matrix elements was obtained by Wishart in 1928. When N > M, what we called the ``Anti-Wishart'' case, the matrix elements are no longer completely independent because the true degrees of freedom becomes smaller than the number of matrix elements. Due to this singular nature, analytical derivation of the probability distribution function is much more involved than the corresponding Wishart case. For a class of general random vectors, we obtain the analytical distribution function in a closed form, which is a product of various factors and delta function constraints, composed of various determinants. The distribution function of the matrix element for the $M\ge N$ case with the same class of random vectors is also obtained as a by-product. Our result is closely related to and should be valuable for the study of random magnet problem and information redundancy problem.Comment: to appear in Physica

The Distribution of Alpha Elements in Ultra-Faint Dwarf Galaxies

The Milky Way ultra-faint dwarf galaxies (UFDs) contain some of the oldest, most metal-poor stars in the Universe. We present [Mg/Fe], [Si/Fe], [Ca/Fe], [Ti/Fe], and mean [alpha/Fe], abundance ratios for 61 individual red giant branch stars across 8 UFDs. This is the largest sample of alpha abundances published to date in galaxies with absolute magnitudes M_V > -8, including the first measurements for Segue 1, Canes Venatici II, Ursa Major I, and Leo T. Abundances were determined via medium-resolution Keck/DEIMOS spectroscopy and spectral synthesis. The sample spans the metallicity range -3.4 < [Fe/H] < -1.1. With the possible exception of Segue 1 and Ursa Major II, the individual UFDs show on average lower [alpha/Fe] at higher metallicities, consistent with enrichment from Type Ia supernovae. Thus even the faintest galaxies have undergone at least a limited level of chemical self-enrichment. Together with recent photometric studies, this suggests that star formation in the UFDs was not a single burst, but instead lasted at least as much as the minimum time delay of the onset of Type Ia supernovae (~100 Myr) and less than ~2 Gyr. We further show that the combined population of UFDs has an [alpha/Fe] abundance pattern that is inconsistent with a flat, Galactic halo-like alpha abundance trend, and is also qualitatively different from that of the more luminous CVn I dSph, which does show a hint of a plateau at very low [Fe/H].Comment: 14 pages, 6 figures, re-submitted to ApJ with revisions based on referee repor