How to form carbon nanoscrolls with non-uniform curvatures is worthy of a detailed investigation. The first-principles method is suitable for studying the combined effects due to the finite-size confinement, the edge-dependent interactions, the interlayer atomic interactions, the mechanical strains, and the magnetic configurations. The complex mechanisms can induce unusual essential properties, e.g., the optimal structures, magnetism, band gaps and energy dispersions. To reach a stable spiral profile, the requirements on the critical nanoribbon width and overlapping length will be thoroughly explored by evaluating the width-dependent scrolling energies. A comparison of formation energy between armchair and zigzag nanoscrolls is useful in understanding the experimental characterizations. The spin-up and spin-down distributions near the zigzag edges are examined for their magnetic environments. This accounts for the conservation or destruction of spin degeneracy. The various curved surfaces on a relaxed nanoscroll will create complicated multi-orbital hybridizations so that the low-lying energy dispersions and energy gaps are expected to be very sensitive to ribbon width, especially for those of armchair systems. Finally, the planar, curved, folded, and scrolled graphene nanoribbons are compared with one another to illustrate the geometry-induced diversity.

This study proposes another computational approach to solve a stochastic attrition model. The initial contact forces for both sides can be treated as a random variable. The approach is manipulated in a matrix form, and on account of the special form of its infinitesimal generator, some recursive algorithms are derived to compute the intended results. Numerical results to illustrate the differences between the proposed model and the stochastic model with known initial contact forces are presented.T3 order to understand more about the combat' dynamics, several authors such as ~ennings'~, hat^, Weale4, Karmeshu and Jaiswa15. and Jaiswa16 have paid attention to the developing of a stochastic combat model recently. Although the stochastic attrition is better in representing the reality of combat attrition phenomena, it is considerably less convenient to handle and compute.In this study, we pkopose another computational approach to solve the stochastic attrition model. The approach is based upon defining a Markov attrition process and applying the concept of matrix-geometric computational algorithms by ~euts'. Since the stochastic model is manipulated in the matrix form and on account of the special form of its infinitesimal generator, some efficient recursive algorithms can be derived. Numerical results by taking the example from ~a i s w a l~ are presented.

A model for a heterogeneous dynamic combat as a continuous-time Markov process has been studied, and on account of the special form of its infinitesimal generator, recursive algorithms are derived to compute the important characteristics of the combat, such as the combat time distribution, expected value and variance, and the probability of winning and expected survivors. Numerical results are also presented. This approach can also be used to consider initial contact forces of both sides as random variables. The other method is modelling the force-on-force attrition as a continuous time Markov process. This approach is better for not only representing the combat phenomena, but also deriving the stochastic behaviours. Feigin et at developed a simple model describing the dominant features of air combat by a continuous time discrete-space Markov process. J aiswal5 , and Chang and Menq6 proposed two different computational methods to derive the results for homogeneous combat forces. Karr7 discussed the analogical extension to the heterogeneous system and concluded that it is sufficiently intractable.