Volume 69, Issue 3-4

The Dirac equation is solved exactly under the condition of spin symmetry for a spin 1=2 particle in the field of Mie-type potential and a Coulomb-like tensor interaction via the Laplace transform approach (LTA). The Dirac bound state energy equation and the corresponding normalized wave functions are obtained in closed forms with any spin-orbit coupling term k. The effects of the tensor interaction and the potential parameters on the bound states are also studied. It is noticed that the tensor interaction removes degeneracy between two states in spin doublets. Some numerical results are given and a few special cases of interest are presented. We have demonstrated that in the nonrelativistic limit, the solutions of the Dirac system converges to that of the Schrödinger system. The nonrelativistic limits of the present solutions are compared with the ones obtained by findings of other methods. Our results are sufficiently accurate for practical purpose.

The reduced partition function ratio (RPFR) of lithium in lithium-graphite intercalation compounds (Li-GICs) was evaluated at the UB3LYP=6-311G(d) level of theory. The partition functions were written in the usual rigid-rotor harmonic oscillator approximation.With a C 24 H 12 coronene molecule as the model of graphene, lithium-coronene sandwich, and club sandwich compounds were considered as models of Li-GICs. The estimated value of the 6 Li-to- 7 Li RPFR was 1.0402 at 25 °C, which yielded 1.034 as the value of the equilibrium constant, K, of the lithium isotope exchange reaction between a lithium ion in an ethylene carbonate=ethylmethyl carbonate mixed solvent and a lithium atom in interlayer space of graphite. The estimated value of K was larger than the experimental value of 1.025. The unsatisfactory agreement between the estimated and experimental K values suggested that larger molecules should be used as models of graphene and that the vibrational anharmonicity should also be taken into consideration.

This presentation provides a mathematical survey of parameters used to adjust weights in Hopfield neural network (HNN). These parameter values are used to adjust the rate of convergence, optimality ratio, and precision of the network for a given problem. Optimal shortest routing path computation is discussed here as one of the optimization problems. A detailed analysis of one such energy function is carried out by varying the parameter values one by one in different ranges. A comprehensive conclusion is presented here about the proper criterion for selecting the values of these effectual parameters.

In this article, the homotopy analysis method is used to obtain approximate analytic solutions of the time-fractional diffusion-wave equation with given initial conditions. A special effort has been given to show the effect of reaction term with long term correlation to the diffusion-wave solutions for various values of anomalous exponent to constitute a good mathematical model useful for various engineering and scientific systems. Effects of parameters on the solution profile are calculated numerically and presented through graphs for different particular cases. Sub-diffusion and super-diffusion phenomena for birth and death processes are also shown through figures.

According to the statistical data, a total of 23 M ≥ 8 earthquakes occurred in Mainland China from 1303 to 2012. The seismic activity of M ≥ 8 earthquakes has showed an obvious self-organized orderliness. It should be remarked especially that there were three ordered pairs of M ≥8 earthquakes occurred in West China during 1902 - 2001, of which the time interval in each pair of two earthquakes was four years. This is a unique and rare earthquake example in earthquake history of China and the world. In the guidance of the information forecasting theory of Wen-Bo Weng, based on previous research results, combining ordered analysis with complex network technology, this paper focuses on the summary of the ordered network structure of M ≥ 8 earthquakes, supplements new information, constructs and further optimizes the 2D- and 3D-ordered network structure of M ≥ 8 earthquakes to make prediction research. At last, a new prediction opinion is presented that the future ordered pair of great earthquakes will probably occur around 2022 and 2026 in Mainland China.

Coupled nonlinear partial differential equations describing the spatio-temporal dynamics of predator-prey systems and nonlinear telegraph equations have been widely applied in many real world problems. So, finding exact solutions of such equations is very helpful in the theories and numerical studies. In this paper, the Kudryashov method is implemented to obtain exact travelling wave solutions of such physical models. Further, graphic illustrations in two and three dimensional plots of some of the obtained solutions are also given to predict their behaviour. The results reveal that the Kudryashov method is very simple, reliable, and effective, and can be used for finding exact solution of many other nonlinear evolution equations.

Approximate analytical solutions of the Dirac equation are obtained for the Hellmann potential, the Wei-Hua potential, and the Varshni potential with any k-value for the cases having the Dirac equation pseudospin and spin symmetries. Closed forms of the energy eigenvalue equations and the spinor wave functions are obtained by using the Nikiforov-Uvarov method and some tables are given to see the dependence of the energy eigenvalues on different quantum number pairs (n;κ).

Since the 1930s, astronomical observations have accumulated evidence that our understanding of the dynamics of galaxies and groups of galaxies is grossly incomplete: assuming the validity of Newton’s law of gravity on astronomical scales, the observed mass (stored in stars and interstellar gas) of stellar systems can account only for roughly 10% of the dynamical (gravitating) mass required to explain the high velocities of stars in those systems. The standard approach to this ‘missing mass problem’ has been the postulate of ‘dark matter’, meaning an additional, electromagnetically dark, matter component that provides the missing mass. However, direct observational evidence for dark matter has not been found to date. More importantly, astronomical observations obtained during the last decade indicate that dark matter cannot explain the kinematics of galaxies. Multiple observations show that the discrepancy between observed and dynamical mass is a function of gravitational acceleration (or field strength) but not of other parameters (size, rotation speed, etc.) of a galaxy; the mass discrepancy appears below a characteristic and universal acceleration aM = (1:1±0:1) · 10 -10 ms -2 (Milgrom’s constant). Consequently, the idea of a modified law of gravity, specifically the ansatz of modified Newtonian dynamics (MOND), is becoming increasingly important in astrophysics. MOND has successfully predicted various important empirical relations of galaxy dynamics, including the famous Tully-Fisher and Faber-Jackson relations. MOND is found to be consistent with stellar dynamics from binary stars to clusters of galaxies, thus covering stellar systems spanning eight orders of magnitude in size and 14 orders of magnitude in mass. These developments have the potential to initiate a paradigm shift from dark matter to a modified law of gravity as the physical mechanism behind the missing mass problem.

The present study is an attempt to find a solution for steady flow of a third-grade fluid by utilizing spectral methods based on rational Christov functions. This problem is described as a nonlinear twopoint boundary value problem. The following method tries to solve the problem on the infinite domain without truncating it to a finite domain and transforms the domain of the problem to a finite domain. Researchers in this try to solve the problem by using anew modified rational Christov functions and normal rational Christov function. Finally, the findings of the current study, i. e., proposal methods, numerical out cames and other methods were compared with each other.