The 1971 Fortuin-Kasteleyn-Ginibre (FKG) inequality for two monotone functions on a distributive lattice is well known and has seen many applications in statistical mechanics other fields of mathematics. In 2008 one us (Sahi) conjectured an extended version this all $n>2$ lattice. Here we prove the conjecture special cases: unit square ${\mathbb R}^k$ whose upper level sets are $k$-dimensional ...

We propose a new single-defect microcavity which supports well-confined whispering gallery mode in octagonal quasiphotonic crystals (OQPC) by shifting the eight nearest air holes around the cavity. Lasing action is observed and compared with the calculated results obtained using the finite-difference time-domain method. We also investigate the threshold dependence on the number of lattice perio...

We report the simulation results for impact of nonlinear Kerr effect on band structures of a ‎two dimensional photonic crystal (2D-PhC) with no defect, a PhC based W1-waveguide ‎‎(W1W), and also Coupled-Cavity Waveguides (CCWs). All PhC structres are assumed to a ‎square lattice of constant a made of GaAs rods of radius r=0.2a, in an air background. The ‎numerical simulation was performed using...

We consider an overdamped Brownian particle, exposed to a two-dimensional, square lattice potential and a rectangular ac drive. Depending on the driving amplitude, the linear response to a weak dc force along a lattice symmetry axis consist in a mobility in basically any direction. In particular, motion exactly opposite to the applied dc force may arise. Upon changing the angle of the dc force ...

For a photonic crystal (PhC) of finite size, it is important to calculate its transmission and reflection spectra. For two-dimensional (2-D) PhCs composed of a square lattice of circular cylinders, the problem can be solved by an efficient method based on the Dirichletto-Neumann (DtN) map of the unit cell and a marching scheme using a pair of operators. In this paper, the DtN operator marching ...

We use a very fast and efficient algorithm to study by Monte Carlo methods the equilibrium cluster distribution C,(L) , the mean number of clusters per lattice site containing I particles in a square lattice of L2 sites, of the two-dimensional fsing model at the critical point. Finite-size scaling theory is then used to analyse the scalingansatz C,(L) = [-If(2'/L), T and s being critical expone...

It is well-known that the question of whether a given finite region can be tiled with a given set of tiles is NP-complete. We show that the same is true for the right tromino and square tetromino on the square lattice, or for the right tromino alone. In the process, we show that Monotone 1-in-3 Satisfiability is NP-complete for planar cubic graphs. In higher dimensions, we show NP-completeness ...

The mapping of phenotype onto genotype for a set of functional model proteins is accomplished by exhaustive enumeration on a three-dimensional diamond lattice. Chains of up to 25 monomers are investigated and their evolution characterized. The model is used to investigate the origins of designability. Highly designable functional model protein structures possess contact maps that have a relativ...

We demonstrate that a Bose-Einstein condensate in an optical lattice forms a reconfigurable matter-wave structure with a band-gap spectrum, which resembles a nonlinear photonic crystal for light waves. We study in detail the case of a two-dimensional square optical lattice and show that this atomic band-gap structure allows nonlinear localization of atomic Bloch waves in the form of two-dimensi...