The present paper aims to systematically investigate whether the differences in the calculations of atomic data for Mg-like S V ion are due to the different treatment of relativistic effects or to the approximation made in solving the resultant scattering equations. We have carried out independent calculations using several procedures for including these effects, ranging from transforming the non-relativistic K-matrix to solving the Dirac equation. We discuss where can accurate results be obtained using the Breit-Pauli Hamiltonian, and when is it necessary to use the Dirac Hamiltonian for this ion. Comparisons with earlier reported data are provided.

Let B-3(K) =3D {v : there is an S(3, K, v)}. For K =3D {4} or {4, 6}, B-3(K) has been determined by Hanani, and for K =3D {4, 5} or {4, 5, 6}, B-3(K) has been determined by the first author. In this paper, we investigate the case K =3D {4, 5, 7}. A necessary condition for v is an element of B-3({4, 5, 7}) is v =3D 1, 2 (mod 3). It is known that B-3({4, 5}) =3D {v >=3D 4 : v =3D 1, 2, 4, 5, 8, 10 (mod 12), v not equal 13} subset of B-3({4, 5, 7}), and that there is an S(3, {4, 5, 7}, v) for all v =3D 7 (mod 12) with a possible exception v =3D 19. We need to consider the case v equivalent to 11 (mod 12). It is proved that there is an S(3, {4, 5, 7}, v) for all v =3D 11 (mod 12) with an exception v equivalent to 11 and a possible exception v =3D 23, thereby, B-3({4, 5, 7}). {19, 23} =3D {v >=3D 4 : v equivalent to 1,2 (mod3), v not equal 11, 13}. c 2016 Wiley Periodicals,

We introduce an approach to estimate the ratio between P- and S-wave velocities, v(p)/v(s) in the scope of elastic full waveform inversion (FWI). Elastic FWI is generally implemented with local optimization methods relying on initial estimates of the long wavelengths of P- and S-wave models. However, successful inversions can be hindered if an accurate enough relation between v(p) and v(s) velocities is not used as a constraint. This relation can be estimated from empirical relations. Herein, we introduce an alternative approach based upon a semi-global inversion scheme. We observe that for a large number of cases, and particularly in the context of FWI, v(p)/v(s) can be represented on a sparse basis. This sparse basis has a much smaller dimension than that of the typical model space in elastic FWI. This creates the possibility of using global optimization methods. The optimal estimate of v(p)/v(s) is obtained with quantum particle swarm optimization (QPSO). This method probes a population of possible models. The assessment of each model of v(p)/v(s) in the population is obtained with nested local iterations updating for v(p) only. Conventional elastic FWI is then carried out for jointly estimating high-resolution models of v(p) and v(s). We demonstrate with synthetic examples that the estimates of v(p) are relatively robust to errors in the estimated v(p)/v(s) and that effectively a sparse representation of the model of v(p)/v(s) is feasible for the reconstruction of a model of v(s). We also demonstrate that the proposed approach performs better than constraining elastic FWI with an empirical relation between v(p) and v(s) leading to improved estimates of models of v(p) and v(s) from seismic data.