Narrow Results

In this paper, we consider the commutative non-unitary ring of order four defined as $I=\{a,b|2a=0,2b=0,a^2=b,ab=0\}.$ Self-orthogonal codes for a symplectic inner product over $I$ are introduced. A mass formula to enumerate them under symplectic equivalence is given. An application is a classification of such codes in short lengths.

Nuclear masses are calculated using the modified Bethe–Weizsäcker mass formula in which the isotonic shifts have been incorporated. The results are compared with the improved liquid drop model with isotonic shift. Mass excesses predicted by this method compares well with the microscopic–macroscopic model while being much more simple. The neutron and proton drip lines have been predicted using this modified Bethe–Weizsäcker mass formula with isotonic shifts.

We derive area product, entropy product, area sum and entropy sum of the event horizon and Cauchy horizons for Kerr–Newman–Taub–NUT (Newman–Unti–Tamburino) black hole in four-dimensional Lorentzian geometry. We observe that these thermodynamic products are not universal (mass-independence) for this black hole (BH), whereas for Kerr–Newman (KN) BH such products are universal (mass-independence). We also examine the entropy sum and area sum. It is shown that they all depend on mass, charge and NUT parameter of the background spacetime. Thus, we can conclude that the universal (mass-independence) behavior of area product and entropy product, area sum and entropy sum for Kerr–Newman–Taub–NUT (KNTN) BH fails and which is also quite different from KN BH. We further show that the KNTN BH do not possess first law of BH thermodynamics and Smarr–Gibbs–Duhem relations, and that such relations are unlikely in the KN case. The failure of these aforementioned features are due to presence of the nontrivial NUT charge which makes the spacetime to be asymptotically non-flat, in contrast with KN BH. The other reason of the failure is that Lorentzian KNTN geometry contains Dirac–Misner type singularity, which is a manifestation of a nontrivial topological twist of the manifold. The BH mass formula and Christodoulou–Ruffini mass formula for KNTN BHs are also derived. Finally, we compute the area bound which is just Penrose like inequality for event horizon. From area bound we derive entropy bound. These thermodynamic products on the multi-horizon play a crucial role in BH thermodynamics to understand the microscopic nature of BH entropy.

A Buchdahl star is a highly compact star for which the boundary radius $R$ obeys $R=\frac{9}{8}{r}_{+}$, where $r_{+}$ is the gravitational radius of the star itself. A quasiblack hole is a maximum compact star, or more generically a maximum compact object, for which the boundary radius $R$ obeys $R=r_{+}$. Quasiblack holes are objects on the verge of becoming black holes. Continued gravitational collapse ends in black holes and has to be handled with the Oppenheimer–Snyder formalism. Quasistatic contraction ends in a quasiblack hole and should be treated with appropriate techniques. Quasiblack holes, not black holes, are the real descendants of Mitchell and Laplace dark stars. Quasiblack holes have many interesting properties. We develop the concept of a quasiblack hole, give several examples of such an object, define what it is, draw its Carter–Penrose diagram, study its pressure properties, obtain its mass formula, derive the entropy of a nonextremal quasiblack hole and through an extremal quasiblack hole give a solution to the puzzling entropy of extremal black holes.

The binding energies per-nucleon for 1654 nuclei, whose mass numbers range from 16 to 263 and charge numbers range from 8 to 106, are calculated by the relativistic mean field theory, with finite nucleon size effect being taken into account. The calculated energy surface goes through the middle of experimental points, and the root mean square deviation for the binding energies per-nucleon is 0.08163 MeV. The numerical results may be well simulated by a droplet model type mass formula. The droplet model is therefore put on the relativistic mean field theoretical foundations.

Proton and neutron separation energies are calculated using the extended Bethe–Weizsäcker mass formula. This modified Bethe–Weizsäcker mass formula describes minutely the positions of all the old and the new magic numbers. It accounts for the disappearance of some traditional magic numbers for neutrons and provides extra stability for some new neutron numbers. The neutron and proton drip lines are predicted using this extended Bethe–Weizsäcker mass formula. The implications of the proton drip line on the astrophysical rp-process and of the neutron drip line on the astrophysical r-process are discussed.

The variation of nuclear shell effects with nucleon numbers is evaluated using the modified Bethe–Weizsäcker mass formula (BWM) and the measured atomic masses. The shell effects at magic neutron numbers N=8, 20, 28, 50, 82 and 126 and magic proton numbers Z=8, 20, 28, 50 and 82 are found to vary rapidly approaching the drip-lines. The shell effect due to one magic number increases on approaching another magic number. Thus, shell effects are not always negligible near the drip-lines.