The Hofstadter problem is studied on the hexagonal lattice. We first establish a relation between the spectra for the hexagonal lattice and for its dual lattice, the triangular lattice. Following the idea of Faddeev and Kashaev, we then obtain the Bethe-Ansatz equations for this system.

We systematically investigate and compare general methods of designing single mode photonic crystal waveguides in a two-dimensional hexagonal lattice of air holes in a dielectric material. We apply the rather general methods to dielectric-core hexagonal lattice photonic crystals since they have not been widely explored before. We show that it is possible to obtain single mode guiding in a limit...

Boron nitride (BN) does not exist as a natural mineral, but can be synthesized in its hexagonal (hBN) and cubic form (cBN). Both phases are stable at ambient conditions and are used in a variety of applications, e.g. as lubricant (hBN) or abrasive (cBN). hBN has a similar layered structure as graphite, whereas cBN is the second-hardest material after diamond with a similar unit cell. There exis...

My general research interests include image processing, numerical analysis, algebra, combinatorics, differential equations, neural networks, pattern recognition, and statistics. I have taken a multitude of courses at the graduate level in mathematics, applied mathematics, computer science and statistics. I am particularly interested in mathematical problems that are driven by applications. Beca...

We produce an explicit parameterization of well-rounded sublattices of the hexagonal lattice in the plane, splitting them into similarity classes. We use this parameterization to study the number, the greatest minimal norm, and the highest signal-to-noise ratio of well-rounded sublattices of the hexagonal lattice of a fixed index. This investigation parallels earlier work by Bernstein, Sloane, ...

We analyze quasi-phase-matched (QPM) conversion efficiency of the five possible types of periodic two-dimensional nonlinear structures: Hexagonal, square, rectangular, centered-rectangular, and oblique. The frequency conversion efficiency, as a function of the two-dimensional quasi-phase-matching order, is determined for the general case. Furthermore, it is demonstrated for two basic feasible m...

A 1-2 model configuration is a subgraph of the hexagonal lattice, in which the edgedegree of each vertex is either 1 or 2. We prove that for any translation invariant Gibbs measure of the 1-2 model configurations on the whole-plane hexagonal lattice, almost surely there are no infinite paths. Using a measure-preserving correspondence between 1-2 model configurations on the hexagonal lattice and...

In this paper, a new structure is provided for the dispersion compensating photonic crystal fibers in order to broaden the chromatic dispersion and increase the dispersion compensating capability in a wide wavelength range. In the structure, putting elliptical holes in the first ring of the inner core clad of a dispersion compensating fiber of the hexagonal lattice, increases the wavelength ran...

In this study, an octagonal-based self-organizing network’s lattice structure is proposed to allow more exploration and exploitation in updating the weights for better mapping and classification performances. The neighborhood of the octagonal-based lattice structure provides more nodes for the weights updating than standard hexagonal-based lattice structure. Based on our experiment, the octagon...