Computationally efficient algorithms for the two-dimensional Kolmogorov-Smirnov test

Goodness-of-fit statistics measure the compatibility of random samples against some theoretical or reference probability distribution function. The classical one-dimensional Kolmogorov-Smirnov test is a non-parametric statistic for comparing two empirical distributions which defines the largest absolute difference between the two cumulative distribution functions as a measure of disagreement. Adapting this test to more than one dimension is a challenge because there are 2^d-1 independent ways of ordering a cumulative distribution function in d dimensions. We discuss Peacock's version of the Kolmogorov-Smirnov test for two-dimensional data sets which computes the differences between cumulative distribution functions in 4n^2 quadrants. We also examine Fasano and Franceschini's variation of Peacock's test, Cooke's algorithm for Peacock's test, and ROOT's version of the two-dimensional Kolmogorov-Smirnov test. We establish a lower-bound limit on the work for computing Peacock's test of Omega(n^2.lg(n)), introducing optimal algorithms for both this and Fasano and Franceschini's test, and show that Cooke's algorithm is not a faithful implementation of Peacock's test. We also discuss and evaluate parallel algorithms for Peacock's test

Combinatorial synthesis of oxysulfides in the lanthanum-bismuth-copper system

Establishing synthesis methods for a target material constitutes a grand challenge in materials research, which is compounded with use-inspired specifications on the format of the material. Solar photochemistry using thin film materials is a promising technology for which many complex materials are being proposed, and the present work describes application of combinatorial methods to explore the synthesis of predicted La–Bi–Cu oxysulfide photocathodes, in particular alloys of LaCuOS and BiCuOS. The variation in concentration of three cations and two anions in thin film materials, and crystallization thereof, is achieved by a combination of reactive sputtering and thermal processes including reactive annealing and rapid thermal processing. Composition and structural characterization establish composition-processing-structure relationships that highlight the breadth of processing conditions required for synthesis of LaCuOS and BiCuOS. The relative irreducibility of La oxides and limited diffusion indicate the need for high temperature processing, which conflicts with the temperature limits for mitigating evaporation of Bi and S. Collectively the results indicate that alloys of these phases will require reactive annealing protocols that are uniquely tailored to each composition, motivating advancement of dynamic processing capabilities to further automate discovery of synthesis routes

Closed trajectories of a particle model on null curves in anti-de Sitter 3-space

We study the existence of closed trajectories of a particle model on null curves in anti-de Sitter 3-space defined by a functional which is linear in the curvature of the particle path. Explicit expressions for the trajectories are found and the existence of infinitely many closed trajectories is proved.Comment: 12 pages, 1 figur

The enhancement of ferromagnetism in uniaxially stressed diluted magnetic semiconductors

We predict a new mechanism of enhancement of ferromagnetic phase transition temperature $T_c$ in uniaxially stressed diluted magnetic semiconductors (DMS) of p-type. Our prediction is based on comparative studies of both Heisenberg (inherent to undistorted DMS with cubic lattice) and Ising (which can be applied to strongly enough stressed DMS) models in a random field approximation permitting to take into account the spatial inhomogeneity of spin-spin interaction. Our calculations of phase diagrams show that area of parameters for existence of DMS-ferromagnetism in Ising model is much larger than that in Heisenberg model.Comment: Accepted for publication in Phys. Rev.

Analyzing machine learning models to accelerate generation of fundamental materials insights

Machine learning for materials science envisions the acceleration of basic science research through automated identification of key data relationships to augment human interpretation and gain scientific understanding. A primary role of scientists is extraction of fundamental knowledge from data, and we demonstrate that this extraction can be accelerated using neural networks via analysis of the trained data model itself rather than its application as a prediction tool. Convolutional neural networks excel at modeling complex data relationships in multi-dimensional parameter spaces, such as that mapped by a combinatorial materials science experiment. Measuring a performance metric in a given materials space provides direct information about (locally) optimal materials but not the underlying materials science that gives rise to the variation in performance. By building a model that predicts performance (in this case photoelectrochemical power generation of a solar fuels photoanode) from materials parameters (in this case composition and Raman signal), subsequent analysis of gradients in the trained model reveals key data relationships that are not readily identified by human inspection or traditional statistical analyses. Human interpretation of these key relationships produces the desired fundamental understanding, demonstrating a framework in which machine learning accelerates data interpretation by leveraging the expertize of the human scientist. We also demonstrate the use of neural network gradient analysis to automate prediction of the directions in parameter space, such as the addition of specific alloying elements, that may increase performance by moving beyond the confines of existing data

Mapping the three-body system - decay time and reversibility

In this paper we carry out a quantitative analysis of the three-body systems and map them as a function of decaying time and intial conguration, look at this problem as an example of a simple deterministic system, and ask to what extent the orbits are really predictable. We have investigated the behavior of about 200 000 general Newtonian three body systems using the simplest initial conditions. Within our resolution these cover all the possible states where the objects are initially at rest and have no angular momentum. We have determined the decay time-scales of the triple systems and show that the distribution of this parameter is fractal in appearance. Some areas that appear stable on large scales exhibit very narrow strips of instability and the overall pattern, dominated by resonances, reminds us of a traditional Maasai warrior shield. Also an attempt is made to recover the original starting conguration of the three bodies by backward integration. We find there are instances where the evolution to the future and to the past lead to different orbits, in spite of time symmetric initial conditions. This implies that even in simple deterministic systems there exists an Arrow of Time.Comment: 8 pages, 9 figures. Accepted for publication in MNRAS. Includes low-resolution figures. High-resolution figures are available as PNG

Suppression of carrier induced ferromagnetism by composition and spin fluctuations in diluted magnetic semiconductors

We suggest an approach to account for spatial (composition) and thermal fluctuations in "disordered" magnetic models (e.g. Heisenberg, Ising) with given spatial dependence of magnetic spin-spin interaction. Our approach is based on introduction of fluctuating molecular field (rather than mean field) acting between the spins. The distribution function of the above field is derived self-consistently. In general case this function is not Gaussian, latter asymptotics occurs only at sufficiently large spins (magnetic ions) concentrations $n_i$. Our approach permits to derive the equation for a critical temperature $T_c$ of ferromagnetic phase transition with respect to the above fluctuations. We apply our theory to the analysis of influence of composition fluctuations on $T_c$ in diluted magnetic semiconductors (DMS) with RKKY indirect spin-spin interaction.Comment: 6 pages, 2 figure

Sharp lines in the absorption edge of EuTe and Pb0.1_{0.1}Eu0.9_{0.9}Te in high magnetic fields

The optical absorption spectra in the region of the \fd transition energies of epitaxial layers of of EuTe and \PbEuTe, grown by molecular beam epitaxy, were studied using circularly polarized light, in the Faraday configuration. Under \sigmam polarization a sharp symmetric absorption line (full width at half-maximum 0.041 eV) emerges at the low energy side of the band-edge absorption, for magnetic fields intensities greater than 6 T. The absorption line shows a huge red shift (35 meV/T) with increasing magnetic fields. The peak position of the absorption line as a function of magnetic field is dominated by the {\em d-f} exchange interaction of the excited electron and the \Euion spins in the lattice. The {\em d-f} exchange interaction energy was estimated to be $J_{df}S=0.15\pm 0.01$ eV. In \PbEuTe the same absorption line is detected, but it is broader, due to alloy disorder, indicating that the excitation is localized within a finite radius. From a comparison of the absorption spectra in EuTe and \PbEuTe the characteristic radius of the excitation is estimated to be $\sim 10$\AA.Comment: Journal of Physics: Condensed Matter (2004, at press