Direct QR factorizations for tall-and-skinny matrices in MapReduce architectures

The QR factorization and the SVD are two fundamental matrix decompositions with applications throughout scientific computing and data analysis. For matrices with many more rows than columns, so-called "tall-and-skinny matrices," there is a numerically stable, efficient, communication-avoiding algorithm for computing the QR factorization. It has been used in traditional high performance computing and grid computing environments. For MapReduce environments, existing methods to compute the QR decomposition use a numerically unstable approach that relies on indirectly computing the Q factor. In the best case, these methods require only two passes over the data. In this paper, we describe how to compute a stable tall-and-skinny QR factorization on a MapReduce architecture in only slightly more than 2 passes over the data. We can compute the SVD with only a small change and no difference in performance. We present a performance comparison between our new direct TSQR method, a standard unstable implementation for MapReduce (Cholesky QR), and the classic stable algorithm implemented for MapReduce (Householder QR). We find that our new stable method has a large performance advantage over the Householder QR method. This holds both in a theoretical performance model as well as in an actual implementation

Stokes-Einstein relation of the liquid metal rubidium and its relationship to changes in the microscopic dynamics with increasing temperature

For liquid rubidium the Stokes-Einstein (SE) relation is well fulfilled near the melting point with an effective hydrodynamic diameter, which agrees well with a value from structural investigations. A wealth of thermodynamic and microscopic data exists for a wide range of temperatures for liquid rubidium and hence it represents a good test bed to challenge the SE relation with rising temperature from an experimental point of view. We performed classical molecular dynamics simulations to complement the existing experimental data using a pseudopotential, which describes perfectly the structure and dynamics of liquid rubidium. The derived SE relation from combining experimental shear viscosity data with simulated diffusion coefficients reveals a weak violation at about 1.3Tmelting ≈ 400 K. The microscopic relaxation dynamics on nearest neighbor distances from neutron spectroscopy demonstrate distinct changes in the amplitude with rising temperature. The derived average relaxation time for density fluctuations on this length scale shows a non-Arrhenius behavior, with a slope change around 1.5Tmelting ≈ 450 K. Combining the simulated macroscopic self-diffusion coefficient with that microscopic average relaxation time, a distinct violation of the SE relation in the same temperature range can be demonstrated. One can conclude that the changes in the collective dynamics, a mirror of the correlated movements of the particles, are at the origin for the violation of the SE relation. The changes in the dynamics can be understood as a transition from a more viscous liquid metal to a more fluid-like liquid above the crossover temperature range of 1.3–1.5 Tmelting. The decay of the amplitude of density fluctuations in liquid aluminium, lead, and rubidium demonstrates a remarkable agreement and points to a universal thermal crossover in the dynamics of liquid metals

A proper fixed functional for four-dimensional Quantum Einstein Gravity

Abstract: Realizing a quantum theory for gravity based on Asymptotic Safety hinges on the existence of a non-Gaussian fixed point of the theory’s renormalization group flow. In this work, we use the functional renormalization group equation for the effective average action to study the fixed point underlying Quantum Einstein Gravity at the functional level including an infinite number of scale-dependent coupling constants. We formulate a list of guiding principles underlying the construction of a partial differential equation encoding the scale-dependence of f(R)-gravity. We show that this equation admits a unique, globally well-defined fixed functional describing the non-Gaussian fixed point at the level of functions of the scalar curvature. This solution is constructed explicitly via a numerical double-shooting method. In the UV, this solution is in good agreement with results from polynomial expansions including a finite number of coupling constants, while it scales proportional to R2, dressed up with non-analytic terms, in the IR. We demonstrate that its structure is mainly governed by the conformal sector of the flow equation. The relation of our work to previous, partial constructions of similar scaling solutions is discussed

Magnetic field splitting of the spin-resonance in CeCoIn5

Neutron scattering in strong magnetic fields is used to show the spin-resonance in superconducting CeCoIn5 (Tc=2.3 K) is a doublet. The underdamped resonance (\hbar \Gamma=0.069 \pm 0.019 meV) Zeeman splits into two modes at E_{\pm}=\hbar \Omega_{0}\pm g\mu_{B} \mu_{0}H with g=0.96 \pm 0.05. A linear extrapolation of the lower peak reaches zero energy at 11.2 \pm 0.5 T, near the critical field for the incommensurate "Q-phase" indicating that the Q-phase is a bose condensate of spin excitons.Comment: 5 pages, 4 figure

Br diffusion in molten NaBr explored by coherent quasielastic neutron scattering

Molten sodium bromide has been investigated by quasielastic neutron scattering focusing on the wave vector range around the first structure factor peak. The linewidth of the scattering function shows a narrowing around the wave number of the structure factor peak, known as deGennes narrowing. In a monatomic system, this narrowing or in the time domain slowing down, has been related to a self-diffusion process of the caged particle. Here we show that this methodology can be applied to the molten alkali halide NaBr. The incoherent scattering from the sodium ions at small wave vectors provides the self-diffusion coefficient of sodium and the dynamics of bromine ions can be studied at wave numbers around the structure factor peak. With input from molecular dynamics simulations on the partial structure factors, diffusion coefficients of the bromine ions can be obtained. These experimentally derived diffusion coefficients are in good agreement with molecular dynamics simulation results. This methodology to extract self-diffusion coefficients from coherent quasielastic neutron scattering is applicable to binary fluids in general when one particle dominates the scattering response at the structure factor maximum.Postprint (author's final draft