Abstract:
Yamakami [2011, Theoret. Comput. Sci.] studies context-free languages with advice functions. Here, the length of an advice is assumed to be the same as that of an input. Let CFL and CFL/n denote the class of all context-free languages and that with advice functions, respectively. We let CFL(2) denote the class of intersections of two context-free languages. An interesting direction of a research is asking how complex CFL(2) is, relative to CFL. Yamakami raised a problem whether there is a CFL-immune set in CFL(2) - CFL/n. The best known so far is that LSPACE - CFL/n has a CFL-immune set, where LSPACE denotes the class of languages recognized in logarithmic-space. We present an affirmative solution to his problem. Two key concepts of our proof are the nested palindrome and Yamakami's swapping lemma. The swapping lemma is applicable to the setting where the pumping lemma (Bar-Hillel's lemma) does not work. Our proof is an example showing how useful the swapping lemma is.

Abstract:
The non energy-weighted Gamow-Teller(GT) sum rule is satisfied in relativistic models, when all nuclear density-dependent terms, including Pauli blocking terms from nucleon-antinucleon excitations, are taken into account in the RPA correlation function. The no-sea approximation is equivalent to this approximation for the giant GT resonance state and satisfies the sum rule, but each of the total $\beta_-$ and $\beta_+$ strengths is different in the two approximations. It is also shown that the energy-weighted sum of the GT strengths for the $\beta_-$ and $\beta_+$ transitions in RPA is equal to the expectation value of the double commutator of the nuclear Hamiltonian with the GT operator, when the expectation value is calculated with the ground state in the mean field approximation. Since the present RPA neglects renormalization of the divergence, however, the energy-weighted strengths outside of the giant GT resonance region become negative. These facts are shown by calculating in an analytic way the GT strengths of nuclear matter.

Abstract:
Effects of the Dirac sea on the excitation energy of the giant monopole states are investigated in an analytic way within the $\sigma-\omega$ model. The excitation energy is determined by the relativistic Landau-Migdal parameters, $F_0$ and $F_1$. Their analytic expressions are derived in the relativistic random phase approximation(RRPA) without the Dirac sea, with the Pauli blocking terms and with the full Dirac sea. It is shown that in the RRPA based on the mean field approximation the Pauli blocking terms should be included in the configuration space, according to the relativistic Landau theory. In the renormalized RRPA, the incompressibility coefficient becomes negative, if N$\nbar$ excitations are neglected.

Abstract:
The neutron spin-orbit density contributes to the nuclear charge density as a relativistic effect. The contribution is enhanced by the effective mass stemming from the Lorentz-scalar potential in relativistic models. This enhancement explains well the difference between the cross sections of elastic electron scattering off $^{40}$Ca and $^{48}$Ca which was not reproduced in non-relativistic models. The spin-orbit density will be examined in more detail in electron scattering off unstable nuclei which would be available in the future.

Abstract:
Relativistic corrections are investigated to the Gamow-Teller(GT) sum rule with respect to the difference between the $\beta_-$ and $\beta_+$ transition strengths in nuclei. Since the sum rule requires the complete set of the nuclear states, the relativistic corrections come from the anti-nucleon degrees of freedom. In the relativistic mean field approximation, the total GT strengths carried by the nucleon sector is quenched by about 12% in nuclear matter, while by about 8% in finite nuclei, compared to the sum rule value. The coupling between the particle-hole states with the nucleon-antinucleon states is also discussed with the relativistic random phase approximation, where the divergence of the response function is renormalized with use of the counter terms in the Lagrangian. It is shown that the approximation to neglect the divergence, like the no-sea approximation extensively used so far, is unphysical, from the sum-rule point of view.

Abstract:
Gamow-Teller (GT) strength in $^{40}$Ar is studied by shell-model calculations with monopole-based universal intearction, which has tensor components of $\pi$\rho$-meson exchanges. Calculated GT strength is found to be consistent with the experimental data obtained by recent ($p, n$) reactions. Neutrino capture cross sections on $^{40}$Ar for solar neutrinos from $^{8}$B are found to be enhanced compared with previous calculations. The reaction cross sections for multipoles other than $0^{+}$ and $1^{+}$ are obtained by random-phase approximation (RPA). Their contributions become important for neutrino energies larger than 50 MeV.

Abstract:
Roles of antinucleon degrees of freedom in the relativistic random phase approximation(RPA) are investigated. The energy-weighted sum of the RPA transition strengths is expressed in terms of the double commutator between the excitation operator and the Hamiltonian, as in nonrelativistic models. The commutator, however, should not be calculated with a usual way in the local field theory, because, otherwise, the sum vanishes. The sum value obtained correctly from the commutator is infinite, owing to the Dirac sea. Most of the previous calculations takes into account only a part of the nucleon-antinucleon states, in order to avoid the divergence problems. As a result, RPA states with negative excitation energy appear, which make the sum value vanish. Moreover, disregarding the divergence changes the sign of nuclear interactions in the RPA equation which describes the coupling of the nucleon particle-hole states with the nucleon-antinucleon states. Indeed, excitation energies of the spurious state and giant monopole states in the no-sea approximation are dominated by those unphysical changes. The baryon current conservation can be described without touching the divergence problems. A schematic model with separable interactions is presented, which makes the structure of the relativistic RPA transparent.

Abstract:
Revisiting the thirty years-old notions of resource-bounded immunity and simplicity, we investigate the structural characteristics of various immunity notions: strong immunity, almost immunity, and hyperimmunity as well as their corresponding simplicity notions. We also study limited immunity and simplicity, called k-immunity and feasible k-immunity, and their simplicity notions. Finally, we propose the k-immune hypothesis as a working hypothesis that guarantees the existence of simple sets in NP.