In this paper energy bands and Berry curvature of graphene was studied. Desired Hamiltonian regarding the next-nearest neighbors obtained by tight binding model. By using the second quantization approach, the transformation matrix is calculated and the Hamiltonian of system is diagonalized. With this Hamiltonian, the band structure and wave function can be calculated. By using calculated wave f...

In this note we show that for any Hamiltonian defined on a symplectic 4-manifold M and any point p ∈ M , there exists a C-close Hamiltonian whose regular energy surface through p is either Anosov or it contains a homoclinic tangency. Our result is based on a general construction of Hamiltonian suspensions for given symplectomorphisms on Poincaré sections already known to yield similar propertie...

We solve the elliptic equations associated with the Hamiltonian and momentum constraints, corresponding to a system composed of two black holes with arbitrary linear and angular momentum. These new solutions are based on a Kerr-Schild spacetime slicing which provides more physically realistic solutions than the initial data based on conformally flat metric/maximal slicing methods. The singulari...

Our main aim is to present a geometrically meaningful formula for the fundamental solutions to a second order sub-elliptic differential equation and to the heat equation associated with a sub-elliptic operator in the sub-Riemannian geometry on the unit sphere S. Our method is based on the Hamiltonian-Jacobi approach, where the corresponding Hamitonian system is solved with mixed boundary condit...

We present a formula for a classical r-matrix of an integrable system obtained by Hamiltonian reduction of some free field theories using pure gauge symmetries. The framework of the reduction is restricted only by the assumption that the respective gauge transformations are Lie group ones. Our formula is in terms of Dirac brackets, and some new observations on these brackets are made. We apply ...

The 3:1 mean-motion resonance of the planar elliptic restricted three body problem (Sun-Jupiter-asteroid) is considered. The double numeric averaging is used to obtain the evolutionary equations which describe the long-term behavior of the asteroid's argument of pericentre and eccentricity. The existence of the adiabatic chaos area in the system's phase space is shown. The 3:1 mean-motion reson...

The bidimensional Duffing oscillator (2DHD) is a parametric Hamiltonian system, integrable by elliptic functions, which includes as particular cases the isotropic oscillator and the classic Hamiltonian Duffing model. The Stark effect, after the KS transformation is used and the parameters are properly chosen, may be studied as a couple of 2DHD. A particular case of 2DHD system has been consider...

We consider an analytic Hamiltonian system with three degrees of freedom and having a family of periodic orbits with a transition stability complex instability We reduce the Hamiltonian to a normal form around a transition periodic orbit and we obtain H Z r R r The analysis of the truncated normal form Z r allows the description of a Hopf bifurcation of D tori However this communication will co...

In recent years it has been trying that with regard to the question of computational complexity of discrete logarithm more strength and less in the elliptic curve than other hard issues, applications such as elliptic curve cryptography, a blind  digital signature method, other methods such as encryption replacement DLP. In this paper, a new blind digital signature scheme based on elliptic curve...