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...

We discuss the relation between space–time diffeomorphisms and gauge transformations in theories of the Yang–Mills type coupled with Einstein’s general relativity. We show that local symmetries of the Hamiltonian and Lagrangian formalisms of these generally covariant gauge systems are equivalent when gauge transformations are required to induce transformations which are projectable under the Le...

We show that classical or quantum mechanics can both be obtained from an infinite-dimensional Hamiltonian system. The procedure consists in postulating mathematical representatives of the physical observables and performing a reduction of the Hamiltonian system (Η, Ω, Φt) by the corresponding phase groups as symmetry groups.

It is shown that a Cayley multigraph over a group G with generating multiset S is integral (i.e., all of its eigenvalues are integers) if S lies in the integral cone over the boolean algebra generated by the normal subgroups of G. The converse holds in the case when G is abelian. This in particular gives an alternative, character-theoretic proof of a theorem of Bridges and Mena (1982). We exten...

Suppose G is a nilpotent, finite group. We show that if {a, b} is any 2-element generating set of G, then the corresponding Cayley digraph −−→ Cay(G; a, b) has a hamiltonian path. This implies that all of the connected Cayley graphs of valence ≤ 4 on G have hamiltonian paths.

This paper presents theorems which establish the existence of horseshoes and Arnold diffusion for nearly integrable Hamiltonian systems associated with Lie groups. The methods are based on our two previous papers, Holmes and Marsden [1982a], [1982b). The two main examples treated here are as follows: 1. A simplified model of the rigid body with attachments. This system has horseshoes (with one ...

The contributions of Sophya Kowalewski to the integrability theory of the equations for the heavy top extend to a larger class of Hamiltonian systems on Lie groups; this paper explains these extensions, and along the way reveals further geometric significance of her work in the theory of elliptic curves. Specifically, in this paper we shall be concerned with the solutions of the following diffe...