Regularity criterion on the energy conservation for the compressible Navier–Stokes equations

This paper concerns the energy conservation for the weak solutions of the compressible Navier–Stokes equations. Assume that the density is positively bounded, we work on the regularity assumption on the gradient of the velocity, and establish a L p –L s type condition for the energy equality to hold in the distributional sense in time. We mention that no regularity assumption on the density derivative is needed any more.

VENM: An Algorithm to Accurately Calculate Neutral Slopes and Gradients

<p>Mesoscale eddies stir along the neutral plane, and the resulting neutral diffusion is a fundamental aspect of subgrid-scale tracer transport in ocean models. Calculating neutral diffusion traditionally involves calculating neutral slopes and three-dimensional tracer gradients. The calculation of the neutral slope traditionally occurs by computing the ratio of the horizontal to vertical locally referenced potential density derivative. However, this approach is problematic in regions of weak vertical stratification, prompting the use of a variety of ad hoc regularization methods that can lead to rather nonphysical dependencies for the resulting neutral tracer gradients.</p><p>Here we introduce VENM; a search algorithm that requires no ad hoc regularization and significantly improves the numerical accuracy of calculating neutral slopes, neutral tracer gradients, and associated neutral diffusive fluxes. We compare and contrast VENM against a more traditional method, using an independent objective neutrality condition combined with estimates of spurious diffusion, heat transport, and water mass transformation rates. VENM is more accurate, both physically and numerically, and should form the basis for future efforts involving neutral diffusion calculations from observations and possibly numerical model simulations.</p>

Correlations among symmetry energy elements in Skyrme models

Motivated by the interrelationships found between the various symmetry energy elements of the energy density functionals (EDF) based on the Skyrme forces, possible correlations among them are explored. A total of 237 Skyrme EDFs are used for this purpose. As some of these EDFs yield values of a few nuclear observables far off from the present acceptable range, studies are done also with a subset of 162 EDFs that comply with a conservative set of constraints on the values of nuclear matter incompressibility coefficient, effective mass of the nucleon and the isovector splitting of effective nucleon masses to see the enhancement of the correlation strength, if any. The curvature parameter [Formula: see text] and the skewness parameter [Formula: see text] of the symmetry energy are found to be very well correlated with the linear combination of the symmetry energy coefficient and its density derivative [Formula: see text]. The isovector splitting of the effective nucleon mass, however, displays a somewhat meaningful correlation with a linear combination of the symmetry energy, its slope and its curvature parameter.

The symmetry energy: Predictions and constraints

After recalling basic phenomenological features of isospin asymmetric nuclear matter, we review predictions for the interaction part of the symmetry energy obtained from different microscopic approaches. The predictions are compared to updated constraints extracted from heavy-ion (HI) reaction observables of a recent GSI experiment. The discussion is extended to the neutron skin thickness in [Formula: see text] and its relation to the density derivative of the symmetry energy. We underline the importance of giving proper consideration to the theoretical uncertainties of microscopic predictions in order to guide phenomenological analyses. In the end, we report briefly on preliminary neutron star calculations based on chiral nuclear forces and outline future plans.

Direct Density Derivative Estimation

Estimating the derivatives of probability density functions is an essential step in statistical data analysis. A naive approach to estimate the derivatives is to first perform density estimation and then compute its derivatives. However, this approach can be unreliable because a good density estimator does not necessarily mean a good density derivative estimator. To cope with this problem, in this letter, we propose a novel method that directly estimates density derivatives without going through density estimation. The proposed method provides computationally efficient estimation for the derivatives of any order on multidimensional data with a hyperparameter tuning method and achieves the optimal parametric convergence rate. We further discuss an extension of the proposed method by applying regularized multitask learning and a general framework for density derivative estimation based on Bregman divergences. Applications of the proposed method to nonparametric Kullback-Leibler divergence approximation and bandwidth matrix selection in kernel density estimation are also explored.