We study the properties of nuclei in the inner crusts of neutron stars based on the Boguta-Bodmer nonlinear model in the relativistic mean-field theory. We carefully determine the surface diffuseness of the nuclei as the density of matter increases. The imaginary time step method is used to solve the Euler-Lagrange equation derived from the variational principle applied to the semiclassical energy density. It is shown that with increasing density, the spherical nuclei become more neutron rich and eventually merge to form a uniform liquid of neutrons, protons, and electrons. We find that the smaller the value of the incompressibility K, the lower the density at which the phase transition to uniform matter occurs. The relativistic extended Thomas-Fermi method is generalized to investigate nonspherical nuclei. Our results show that the spherical nucleus phase is the only equilibrium state in the inner crusts of neutron stars.

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We consider the problem of identifying the graph topology from a set of smooth graph signals. A well-known approach to this problem is minimizing the Dirichlet energy accompanied with some Frobenius norm regularization. Recent works have incorporated the logarithmic barrier on the node degrees to improve the overall graph connectivity without compromising graph sparsity, which is shown to be quite effective in enhancing the quality of the learned graphs. Although a primal-dual algorithm has been proposed in the literature to solve this type of graph learning formulations, it lacks a rigorous convergence analysis and appears to have a slow empirical performance. In this paper, we cast the graph learning formulation as a nonsmooth, strictly convex optimization problem and develop an efficient alternating direction method of multipliers to solve it. We show that our algorithm converges to the global minimum with arbitrary initialization. We conduct extensive experiments on various synthetic and realworld graphs, the results of which show that our method exhibits sharp linear convergence and is substantially faster than the commonly adopted primal-dual method.

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