The problem of enumerating spanning trees in lattices with Klein bottle boundary condition is considered here. The exact closed-form expressions of the numbers of spanning trees for 4.8.8 lattice, hexagonal lattice, and 3(3) · 4(2) lattice on the Klein bottle are presented.

Hubbard-type models on the hexagonal lattice are of great interest, as they provide realistic descriptions graphene and other related materials. Hybrid Monte Carlo simulations offer a first-principles approach to study their phase structure. Here, we review present status our work in this direction.

Spatial resonances leading to superlattice hexagonal patterns, known as "black-eyes," and superposition patterns combining stripes and/or spots are studied in a reaction-diffusion model of two interacting Turing modes with different wavelengths. A three-phase oscillatory interlacing hexagonal lattice pattern is also found, and its appearance is attributed to resonance between a Turing mode and ...

TEM analysis of mesoporous materials is generally undertaken to give qualitative results. Accurate quantitative analysis is demonstrated in this study. A systematic image analysis of a powder form of a hexagonal mesoporous material known as KIT-6 is conducted using a transmission electron microscope (TEM). Three types of image contrast typically appear in this material (a hexagonal honeycomb st...