Electrical conductivity of the quark-gluon plasma: perspective from lattice QCD

A discussion on the electrical conductivity of the quark-gluon plasma as determined by lattice QCD is given. After a reminder of basic definitions and expectations, various methods for spectral reconstruction are reviewed, including the use of Ansätze and sum rules, the Maximum Entropy and Backus-Gilbert methods, and Tikhonov regularisation. A comprehensive overview of lattice QCD results obtained so far is given, including a comparison of the different lattice formulations. A noticeable consistency for the conductivities obtained is seen, in spite of the differences in the lattice setups and spectral reconstruction methods. It is found that in the case of quenched QCD little temperature dependence of $$\sigma /T$$ σ / T is seen in the temperature range investigated, while for QCD with dynamical quarks a reduction of $$\sigma /T$$ σ / T in the vicinity of the thermal crossover is observed, compared to its value in the QGP. Several open questions are posed at the end.

Challenges of Adjusting Single-Nucleotide Polymorphism Effect Sizes for Linkage Disequilibrium

<b><i>Background:</i></b> Genome-wide association studies (GWAS) were successful in identifying SNPs showing association with disease, but their individual effect sizes are small and require large sample sizes to achieve statistical significance. Methods of post-GWAS analysis, including gene-based, gene-set and polygenic risk scores, combine the SNP effect sizes in an attempt to boost the power of the analyses. To avoid giving undue weight to SNPs in linkage disequilibrium (LD), the LD needs to be taken into account in these analyses. <b><i>Objectives:</i></b> We review methods that attempt to adjust the effect sizes (β<i>-</i>coefficients) of summary statistics, instead of simple LD pruning. <b><i>Methods:</i></b> We subject LD adjustment approaches to a mathematical analysis, recognising Tikhonov regularisation as a framework for comparison. <b><i>Results:</i></b> Observing the similarity of the processes involved with the more straightforward Tikhonov-regularised ordinary least squares estimate for multivariate regression coefficients, we note that current methods based on a Bayesian model for the effect sizes effectively provide an implicit choice of the regularisation parameter, which is convenient, but at the price of reduced transparency and, especially in smaller LD blocks, a risk of incomplete LD correction. <b><i>Conclusions:</i></b> There is no simple answer to the question which method is best, but where interpretability of the LD adjustment is essential, as in research aiming at identifying the genomic aetiology of disorders, our study suggests that a more direct choice of mild regularisation in the correction of effect sizes may be preferable.

Energy Resolved Neutron Imaging for Strain Reconstruction Using the Finite Element Method

A novel pulsed neutron imaging technique based on the finite element method is used to reconstruct the residual strain within a polycrystalline material from Bragg edge strain images. This technique offers the possibility of a nondestructive analysis of strain fields with a high spatial resolution. The finite element approach used to reconstruct the strain uses the least square method constrained by the conditions of equilibrium. This inclusion of equilibrium makes the problem well-posed. The procedure is developed and verified by validating for a cantilevered beam problem. It is subsequently demonstrated by reconstructing the strain from experimental data for a ring-and-plug sample, measured at the spallation neutron source RADEN at J-PARC in Japan. The reconstruction is validated by comparison with conventional constant wavelength strain measurements on the KOWARI diffractometer at ANSTO in Australia. It is also shown that the addition of a Tikhonov regularisation scheme further improves the reconstruction.

The inverse problem of recovering an unsteady linear load for an elastic rod of finite length

The main purpose of the paper is to obtain solutions for new non-stationary inverse problems for elastic rods. The objective of this study is to develop and implement new methods, approaches and algorithms for solving non-stationary inverse problems of rod mechanics. The direct non-stationary problem for an elastic rod consists in determining elastic displacements, which satisfies a given equation of non-stationary oscillations in partial derivatives and some given initial and boundary conditions. The solution of inverse retrospective problems with a completely unknown space-time law of load distribution is based on the method of influence functions. With its application, the inverse retrospective problem is reduced to solving a system of integral equations of the Volterra type of the first kind in time with respect to the sought external axial load of the elastic rod. To solve it, the method of mechanical quadratures is used in combination with the Tikhonov regularisation method.

Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems

This paper considers the classical separable nonlinear least squares problem. Such problems can be expressed as a linear combination of nonlinear functions, and both linear and nonlinear parameters are to be estimated. Among the existing results, ill-conditioned problems are less often considered. Hence, this paper focuses on an algorithm for ill-conditioned problems. In the proposed linear parameter estimation process, the sensitivity of the model to disturbance is reduced using Tikhonov regularisation. The Levenberg–Marquardt algorithm is used to estimate the nonlinear parameters. The Jacobian matrix required by LM is calculated by the Golub and Pereyra, Kaufman, and Ruano methods. Combining the nonlinear and linear parameter estimation methods, three estimation models are obtained and the feasibility and stability of the model estimation are demonstrated. The model is validated by simulation data and real data. The experimental results also illustrate the feasibility and stability of the model.

Gaussian process models—II. Lessons for discrete inversion

Summary By starting from a general framework for probabilistic continuous inversion (developed in Part I) and introducing discrete basis functions, we obtain the well-known algorithms for probabilistic least-squares inversion set out by Tarantola & Valette (1982). In doing so, we establish a direct equivalence between the spatial covariance function that must be specified in continuous inversion, and the combination of basis functions and prior covariance matrix that must be chosen for discretised inversion. We show that the common choice of Tikhonov regularisation ($\mathbf {C_m^{-1}} = \sigma ^2\mathbf {I}$) arises from a delta-function spatial covariance, and that this lies behind many of the artefacts commonly associated with discretised inversion. We show that other choices of spatial covariance function can be used to generate regularisation matrices yielding substantially better results, and permitting localisation of features even if global basis functions are employed. We are also able to offer a straightforward explanation for the spectral leakage problem identified by Trampert & Snieder (1996).

Starspot activity of HD 199178

Context. Studying the spots of late-type stars is crucial for distinguishing between the various proposed dynamo mechanisms believed to be the main cause of starspot activity. For this research it is important to collect observation time series that are long enough to unravel both long- and short-term spot evolution. Doppler imaging is a very efficient method for studying spots of stars that cannot be angularly resolved. Aims. High-resolution spectral observations during 1994–2017 are analysed in order to reveal long- and short-term changes in the spot activity of the FK Comae-type subgiant HD 199178. Methods. Most of the observations were collected with the Nordic Optical Telescope. The Doppler imaging temperature maps were calculated using an inversion technique based on Tikhonov regularisation and utilising multiple spectral lines. Results. We present a unique series of 41 temperature maps spanning more than 23 years. All reliable images show a large cool spot region centred near the visible rotation pole. Some lower latitude cool features are also recovered, although the reliability of these is questionable. There is an expected anti-correlation between the mean surface temperature and the spot coverage. Using the Doppler images, we construct the equivalent of a solar butterfly diagram for HD 199178. Conclusions. HD 199178 clearly has a long-term large and cool spot structure at the rotational pole. This spot structure dominated the spot activity during the years 1994–2017. The size and position of the structure has evolved with time, with a gradual increase during the last years. The lack of lower latitude features prevents the determination of a possible differential rotation.