Image processing has traditionally involved the use of square operators on regular rectangular image lattices. For many years the concept of using hexagonal pixels for image capture has been investigated, and several advantages of such an approach have been highlighted. We present a design procedure for hexagonal gradient operators, developed within the finite element framework, for use on hexa...

In this paper, the connection between triangular number , square number for construction and counting the lattice number in lattice vector quantization is presented. We also observe that unique hexagonal lattice can be formed from two triangular numbers and a square number combination. Furthermore, the number of multistage lattice is derived from lattice number . Analysis of the number of latti...

The localization properties of the wave functions of vibrations in two-dimensional (2D) crystals are studied numerically for square and hexagonal lattices within the framework of an algebraic model. The wave functions of 2D lattices have remarkable localization properties, especially at the van Hove singularities (vHs). Finite-size sheets with a hexagonal lattice (graphene-like materials), in a...

We characterize the macroscopic effective mechanical behavior of a graphene sheet modeled by a hexagonal lattice with twoand three-point atomic interactions, using Γ-convergence.

A generalized form of Ginzburg–Landau theory is proposed which explains the non-hexagonal flux-line lattice found both in metallic and magnetic superconductors without invoking any anisotropic material-dependent properties. The Gibbs energy density postulated for magnetic superconductors (g) is of the form g(B, T ) = α|ψ |2 + 1 2β|ψ |4 + [1/(2m)]|(−ih̄∇−2eA)ψ |2 + ∫ (B/μ0 − Mions) · dB − (B/μ0 −...

We characterize the macroscopic effective mechanical behavior of a graphene sheet modeled by a hexagonal lattice of elastic bars, using Γ-convergence.

The luminal and discoid vacuole membranes of the superficial cell layer of the transitional epithelium of the mammalian urinary bladder have been studied by thin-sectioning and freeze-fracture-etch (FFE) electron microscope methods. For the FFE studies membranes were deposited on a cationized glass surface, covered by a thin copper disc, and fractured under liquid N2. Specimens were etched at -...

Motivated by the frequency assignment problem we study the ddistant coloring of the vertices of an infinite plane hexagonal lattice H. Let d be a positive integer. A d-distant coloring of the lattice H is a coloring of the vertices of H such that each pair of vertices distance at most d apart have different colors. The d-distant chromatic number of H, denoted χd(H), is the minimum number of col...

Schmutz Schaller’s conjecture regarding the lengths of the hexagonal versus the lengths of the square lattice is shown to be true. The proof makes use of results from (computational) prime number theory. Using an identity due to Selberg, it is shown that, in principle, the conjecture can be resolved without using computational prime number theory. By our approach, however, this would require a ...