Shear mixing in stellar radiative zones I. Effect of thermal diffusion and chemical stratification

Turbulent transport of chemical elements in radiative zones of stars is considered in current stellar evolution codes thanks to phenomenologically derived diffusion coefficients. Recent local numerical simulations (Prat & Ligni\`eres 2013, A&A, 551, L3) suggest that the coefficient for radial turbulent diffusion due to radial differential rotation satisfies $D_{\rm t}\simeq0.058\kappa/Ri$, in qualitative agreement with Zahn's model. However, this model does not apply when differential rotation is strong with respect to stable thermal stratification or when chemical stratification has a significant dynamical effect, a situation encountered at the outer boundary of nuclear-burning convective cores. We extend our numerical study to consider the effects of chemical stratification and of strong shear, and compare the results with prescriptions used in stellar evolution codes. We performed local, direct numerical simulations of stably stratified, homogeneous, sheared turbulence in the Boussinesq approximation. The regime of high thermal diffusivities, typical of stellar radiative zones, is reached thanks to the so-called small-P\'eclet-number approximation, which is an asymptotic development of the Boussinesq equations in this regime. The dependence of the diffusion coefficient on chemical stratification was explored in this approximation. Maeder's extension of Zahn's model in the strong-shear regime is not supported by our results, which are better described by a model found in the geophysical literature. As regards the effect of chemical stratification, our quantitative estimate of the diffusion coefficient as a function of the mean gradient of mean molecular weight leads to the formula $D_{\rm t}\simeq 0.45\kappa(0.12-Ri_\mu)/Ri$, which is compatible in the weak-shear regime with the model of Maeder & Meynet (1996, A&A, 313, 140).Comment: 10 pages, 9 figures, accepted in A&

Performance analysis for a class of robust adaptive beamformers

Robust adaptive beamforming is a key issue in array applications where there exist uncertainties about the steering vector of interest. Diagonal loading is one of the most popular techniques to improve robustness. Recently, worst-case approaches which consist of protecting the array's response in an ellipsoid centered around the nominal steering vector have been proposed. They amount to generalized (i.e. non necessarily diagonal) loading of the covariance matrix. In this paper, we present a theoretical analysis of the signal to interference plus noise ratio (SINR) for this class of robust beamformers, in the presence of random steering vector errors. A closed-form expression for the SINR is derived which is shown to accurately predict the SINR obtained in simulations. This theoretical formula is valid for any loading matrix. It provides insights into the influence of the loading matrix and can serve as a helpful guide to select it. Finally, the analysis enables us to predict the level of uncertainties up to which robust beamformers are effective and then depart from the optimal SINR

Performance analysis of beamformers using generalized loading of the covariance matrix in the presence of random steering vector errors

Robust adaptive beamforming is a key issue in array applications where there exist uncertainties about the steering vector of interest. Diagonal loading is one of the most popular techniques to improve robustness. In this paper, we present a theoretical analysis of the signal-to-interference-plus-noise ratio (SINR) for the class of beamformers based on generalized (i.e., not necessarily diagonal) loading of the covariance matrix in the presence of random steering vector errors. A closed-form expression for the SINR is derived that is shown to accurately predict the SINR obtained in simulations. This theoretical formula is valid for any loading matrix. It provides insights into the influence of the loading matrix and can serve as a helpful guide to select it. Finally, the analysis enables us to predict the level of uncertainties up to which robust beamformers are effective and then depart from the optimal SINR

Steering vector errors and diagonal loading

Diagonal loading is one of the most widely used and effective methods to improve robustness of adaptive beamformers. In this paper, we consider its application to the case of steering vector errors, i.e. when there exists a mismatch between the actual steering vector of interest and the presumed one. More precisely, we address the problem of optimally selecting the loading level with a view to maximise the signal to interference plus noise ratio in the presence of random steering vector errors. First, we derive an expression for the optimal loading for a given steering vector error and we show that this loading is negative. Next, this optimal loading is averaged with respect to the probability density function of the steering vector errors, yielding a very simple expression for the average optimal loading. Numerical simulations attest to the validity of the analysis and show that diagonal loading with the optimal loading factor derived herein provides a performance close to optimum

Recognizing a relatively hyperbolic group by its Dehn fillings

Dehn fillings for relatively hyperbolic groups generalize the topological Dehn surgery on a non-compact hyperbolic $3$-manifold such as a hyperbolic knot complement. We prove a rigidity result saying that if two non-elementary relatively hyperbolic groups without suitable splittings have sufficiently many isomorphic Dehn fillings, then these groups are in fact isomorphic. Our main application is a solution to the isomorphism problem in the class of non-elementary relatively hyperbolic groups with residually finite parabolic groups and with no suitable splittings.Comment: Minor modification (including typesetting). 56 page

The isomorphism problem for all hyperbolic groups

We give a solution to Dehn's isomorphism problem for the class of all hyperbolic groups, possibly with torsion. We also prove a relative version for groups with peripheral structures. As a corollary, we give a uniform solution to Whitehead's problem asking whether two tuples of elements of a hyperbolic group $G$ are in the same orbit under the action of \Aut(G). We also get an algorithm computing a generating set of the group of automorphisms of a hyperbolic group preserving a peripheral structure.Comment: 71 pages, 4 figure

Matched subspace detection with hypothesis dependent noise power

We consider the problem of detecting a subspace signal in white Gaussian noise when the noise power may be different under the null hypothesis—where it is assumed to be known—and the alternative hypothesis. This situation occurs when the presence of the signal of interest (SOI) triggers an increase in the noise power. Accordingly, it may be relevant in the case of a mismatch between the actual SOI subspace and its presumed value, resulting in a modelling error. We derive the generalized likelihood ratio test (GLRT) for the problem at hand and contrast it with the GLRT which assumes known and equal noise power under the two hypotheses. A performance analysis is carried out and the distributions of the two test statistics are derived. From this analysis, we discuss the differences between the two detectors and provide explanations for the improved performance of the new detector. Numerical simulations attest to the validity of the analysis

Multiple imputation for continuous variables using a Bayesian principal component analysis

We propose a multiple imputation method based on principal component analysis (PCA) to deal with incomplete continuous data. To reflect the uncertainty of the parameters from one imputation to the next, we use a Bayesian treatment of the PCA model. Using a simulation study and real data sets, the method is compared to two classical approaches: multiple imputation based on joint modelling and on fully conditional modelling. Contrary to the others, the proposed method can be easily used on data sets where the number of individuals is less than the number of variables and when the variables are highly correlated. In addition, it provides unbiased point estimates of quantities of interest, such as an expectation, a regression coefficient or a correlation coefficient, with a smaller mean squared error. Furthermore, the widths of the confidence intervals built for the quantities of interest are often smaller whilst ensuring a valid coverage.Comment: 16 page

Matched direction detectors

In this paper, we address the problem of detecting a signal whose associated spatial signature is subject to uncertainties, in the presence of subspace interference and broadband noise, and using multiple snapshots from an array of sensors. To account for steering vector uncertainties, we assume that the spatial signature of interest lies in a given linear subspace H while its coordinates in this subspace are unknown. The generalized likelihood ratio test (GLRT) for the problem at hand is formulated. We show that the GLRT amounts to searching for the best direction in the subspace H after projecting out the interferences. The distribution of the GRLT under both hypotheses is derived and numerical simulations illustrate its performance