SKYRME

Skyrme, T. H., Pyoc. Roy. Soc. [London) A239, 399, "Collective motion in quantum mechanics."... 

Skyrme D. Knowledge Networking Creating the Collaborative Company. Butterworth-Heinemann, 1999. 

At present, antioxidants are extensively studied as supplements for the treatment diabetic patients. Several clinical trials have been carried out with vitamin E. In 1991, Ceriello et al. [136] showed that supplementation of vitamin E to insulin-requiring diabetic patients reduced protein glycosylation without changing plasma glucose, probably due to the inhibition of the Maillard reaction. Then, Paolisso et al. [137] found that vitamin E decreased glucose level and improved insulin action in noninsulin-dependent diabetic patients. Recently, Jain et al. [138] showed that vitamin E supplementation increased glutathione level and diminished lipid peroxidation and HbAi level in erythrocytes of type 1 diabetic children. Similarly, Skyrme-Jones et al. [139] demonstrated that vitamin E supplementation improved endothelial vasodilator function in type 1 diabetic children supposedly due to the suppression of LDL oxidation. Devaraj et al. [140] used the urinary F2-isoprostane test for the estimate of LDL oxidation in type 2 diabetics. They also found that LDL oxidation decreased after vitamin E supplementation to patients. 

RA Skyrme-Jones, RC O Brien, KL Berry, IT Meredith. J Am Coll Cardiol 36 94-102, 2000. 

A description of nuclear matter as an ideal mixture of protons and neutrons, possibly in (5 equilibrium with electrons and neutrinos, is not sufficient to give a realistic description of dense matter. The account of the interaction between the nucleons can be performed in different ways. For instance we have effective nucleon-nucleon interactions, which reproduce empirical two-nucleon data, e.g. the PARIS and the BONN potential. On the other hand we have effective interactions like the Skyrme interaction, which are able to reproduce nuclear data within the mean-field approximation. The most advanced description is given by the Walecka model, which is based on a relativistic Lagrangian and models the nucleon-nucleon interactions by coupling to effective meson fields. Within the relativistic mean-field approximation, quasi-particles are introduced, which can be parameterized by a self-energy shift and an effective mass. 

Recently in applying the non-relativistic Skyrme Hartree-Fock (SHF) model Brown [19] noted that certain combinations of parameters in the SHF are not well determined by a fit to ground state binding energies alone as a result a wide range of predictions for the EoS for PNM can be obtained. At the same time he found a correlation between the derivative of the neutron star EoS (i.e., basically the symmetry pressure po) and the neutron skin in 208Pb. 

Subsequently Furnstahl [20] in a more extensive study pointed out that within the framework of mean field models (both non-relativistic Skyrme as well as relativistic models) there exists an almost linear empirical correlation between theoretical predictions for both 04 and its density dependence, po, and the neutron skin, All lln — Rp, in heavy nuclei. This is illustrated for 208Pb in Fig. 5 (from ref.[20] a similar correlation is found between All and po). Note that whereas the Skyrme results cover a wide range of All values the RMF predictions in general lead to AR > 0.20 fm. 

Figure 5. Neutron skin thickness versus <24 for 208Pb for a variety of mean field models (from [20]). The circles correspond to results for the Skyrme force, the squares to RMF models with mesons, and the triangles to RMF with point couplings the shaded area indicates the range of NR values consistent with the present empirical information for 208Pb.

In the solitonic sector, the CFL chiral Lagrangian [27, 28] gives us the scaling behavior of the coefficient of the Skyrme term and thus shows that the mass of the soliton is of the order of... 

The low energy effective theory supports solitonic excitations which can be identified with the baryonic sector of the theory at non-zero chemical potential. In order to obtain classically stable configurations, it is necessary to include at least a four-derivative term (containing two temporal derivatives) in addition to the usual two-derivative term. Such a term is the Skyrme term ... 

The Wess-Zumino term in Eq. (11) guarantees the correct quantization of the soliton as a spin 1/2 object. Here we neglect the breaking of Lorentz symmetries, irrelevant to our discussion. The Euler-Lagrangian equations of motion for the classical, time independent, chiral field Uo(r) are highly non-linear partial differential equations. To simplify these equations Skyrme adopted the hedgehog ansatz which, suitably generalized for the three flavor case, reads [40] ... 

In atomic nuclei, SRPA was derived [9,10,19] for the demanding Skyrme functional involving a variety of densities and currents (see [20] for the recent review on Skyrme forces). SRPA calculations for isoscalar and isovector giant resonances (nuclear counterparts of electronic plasmons) in doubly magic nuclei demonstrated high accuracy of the method [10]. 

The paper is organized as follows. In Section 2, derivation of the the SRPA formalism is done. Relations of SRPA with other alternative approaches are commented. In Sec. 3, the method to calculate SRPA strength function (counterpart of the linear response theory) is outlined. In Section 4, the particular SRPA versions for the electronic Kohn-Sham and nuclear Skyrme functionals are specified and the origin and role of time-odd currents in functionals are scrutinized. In Sec. 5, the practical SRPA realization is discussed. Some examples demonstrating accuracy of the method in atomic clusters and nuclei are presented. The summary is done in Sec. 6. In Appendix A, densities and currents for Skyrme functional are listed. In Appendix B, the optimal ways to calculate SRPA basic values are discussed. 

Nuclear interaction is very complicated and its explicit form is still unknown. So, in practice different approximations to nuclear interaction are used. Skyrme forces [11,12] represent one of the most successful approximations where the interaction is maximally simplified and, at the same time, allows to get accurate and universal description of both ground state properties and dynamics of atomic nuclei (see [20] a for recent review). Skyrme forces are contact, i.e. 5(fi — 2), which minimizes the computational effort. In spite of this dramatic simplification, Skyrme forcese well reproduce properties of most spherical and deformed nuclei as well as characteristics of nuclear matter and neutron stars. Additional advantage of the Skyrme interaction is that its parameters are directly related to the basic nuclear... 

Although Skyrme forces are relatively simple, they are still much more demanding than the Coulomb interaction. In particular, they deal with a variety of diverse densities and currents. The Skyrme functional reads [12,36,37]... 

Comparison of the Kohn-Sham and Skyrme functionals leads to a natural question why these two functionals exploit, for the time-dependent problem, so different sets of basic densities and currents If the Kohn-Sham functional is content with one density, the Skyrme forces operate with a diverse set of densities and currents, both T-even and T-odd. Then, should we consider T-odd densities as genuine for the description of dynamics of finite many-body systems or they are a pequliarity of nuclear forces This question is very nontrivial and still poorly studied. We present below some comments which, at least partly, clarify this point. 

Actual nuclear forces are of a finite range. These are, for example, Gogny forces [40] representing more realistic approximation of actual nuclear forces than Skyrme approximation. Gogny interaction has no any velocity dependence and fulfills the Galilean invariance. Instead, two-body Skyrme interaction depends on relative velocities k = j2i (Vi — V2), which just simulates the finite range effects [20]. 

The static Hartree-Fock problem assumes T-reversal invariance and T-even single-particle density matrix. In this case, Skyrme forces can be limited by only T-even densities psif) Tsif) In the case of dy-... 

As compared with the Kohn-Sham functional for electronic systems, the nuclear Skyrme functional is less genuine. The main (Coulomb) interaction in the Kohn-Sham problem is well known and only exchange and corellations should be modeled. Instead, in the nuclear case, even the basic interaction is unknown and should be approximated, e.g. by the simple contact interaction in Skyrme forces. 

The crudeness of Skyrme forces has certain consequences. For example, the Skyrme functional has no any exchange-correlation term since the relevant effects are supposed to be already included into numerous Skyrme fitting parameters. Besides, the Skyrme functional may accept a diverse set of T-even and T-dd densities and currents. One may say that T-odd densities appear in the Skyrme functional partly because of its specific construction. Indeed, other effective nuclear forces (Gogny [40], Landau-Migdal [41]) do not exploit T-odd densities and currents for description of nuclear dynamics. 

Implementation of a variety of densities and currents in the Skyrme fuc-tional has, however, some advantages. It is known that different projectiles and external fields used to generate collective modes in nuclear reactions... 

The particular SRPA versions for electronic Kohn-Sham and nuclear Skyrme functional were considered and examples of the calculations for the dipole plasmon in atomic clusters and giant resonances in atomic nuclei were presented. SRPA was compared with alternative methods, in particular with EOM-CC. It would be interesting to combine advantages of SRPA and couled-cluster approach in one powerful method. 

In Skyrme forces, the complete set of the densities involves the ordinary density, kinetic-energy density, spin-orbital density, current density, spin... 

Skyrms, B. (1980) Causal Necessity", New Haven, Conn Yale... 

Some conjectural solutions of many extremal problem on the sphere (Thomson, Tammes, Skyrme problems, etc.) have (solving exactly the problem is almost impossible) the combinatorial structure of GCkfiDodecahedron) or its dual (see [HaS196]). 

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