Geometric frustration of hard-disk packings on cones.

Conical surfaces pose an interesting challenge to crystal growth: A crystal growing on a cone can wrap around and meet itself at different radii. We use a disk-packing algorithm to investigate how this closure constraint can geometrically frustrate the growth of single crystals on cones with small opening angles. By varying the crystal seed orientation and cone angle, we find that-except at special commensurate cone angles-crystals typically form a seam that runs along the axial direction of the cone, while near the tip, a disordered particle packing forms.

Defect absorption and emission for p-atic liquid crystals on cones.

We investigate the ground-state configurations of two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on fixed curved surfaces. We focus on the intrinsic geometry and show that isothermal coordinates are particularly convenient as they explicitly encode a geometric contribution to the elastic potential. In the special case of a cone with half-angle β, the apex develops an effective topological charge of -χ, where 2πχ=2π(1-sinβ) is the deficit angle of the cone, and a topological defect of charge σ behaves as if it had an effective topological charge Q_{eff}=(σ-σ^{2}/2) when interacting with the apex.

Fractional defect charges in liquid crystals with p-fold rotational symmetry on cones.

Conical surfaces, with a δ function of Gaussian curvature at the apex, are perhaps the simplest example of geometric frustration. We study two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on the surfaces of cones. For free boundary conditions at the base, we find both the ground state(s) and a discrete ladder of metastable states as a function of both the cone angle and the liquid crystal symmetry p.

Bottlenecks, Modularity, and the Neural Control of Behavior.

In almost all animals, the transfer of information from the brain to the motor circuitry is facilitated by a relatively small number of neurons, leading to a constraint on the amount of information that can be transmitted. Our knowledge of how animals encode information through this pathway, and the consequences of this encoding, however, is limited. In this study, we use a simple feed-forward neural network to investigate the consequences of having such a bottleneck and identify aspects of the network architecture that enable robust information transfer.

Phonon eigenfunctions of inhomogeneous lattices: Can you hear the shape of a cone?

We study the phonon modes of interacting particles on the surface of a truncated cone resting on a plane subject to gravity, inspired by recent colloidal experiments. We derive the ground-state configuration of the particles under gravitational pressure in the small-cone-angle limit and find an inhomogeneous triangular lattice with spatially varying density but robust local order. The inhomogeneity has striking effects on the normal modes such that an important feature of the cone geometry, namely its apex angle, can be extracted from the lattice excitations.

Statistical mechanics of dislocation pileups in two dimensions.

Dislocation pileups directly impact the material properties of crystalline solids through the arrangement and collective motion of interacting dislocations. We study the statistical mechanics of these ordered defect structures embedded in two-dimensional crystals, where the dislocations themselves form one-dimensional lattices. In particular, pileups exemplify a new class of inhomogeneous crystals characterized by spatially varying lattice spacings.

Statistical Mechanics of Low Angle Grain Boundaries in Two Dimensions.

We explore order in low angle grain boundaries (LAGBs) embedded in a two-dimensional crystal at thermal equilibrium. Symmetric LAGBs subject to a Peierls potential undergo, with increasing temperatures, a thermal depinning transition; above which, the LAGB exhibits transverse fluctuations that grow logarithmically with interdislocation distance. Longitudinal fluctuations lead to a series of melting transitions marked by the sequential disappearance of diverging algebraic Bragg peaks with universal critical exponents.

Eigenvalue repulsion and eigenvector localization in sparse non-Hermitian random matrices.

Complex networks with directed, local interactions are ubiquitous in nature and often occur with probabilistic connections due to both intrinsic stochasticity and disordered environments. Sparse non-Hermitian random matrices arise naturally in this context and are key to describing statistical properties of the nonequilibrium dynamics that emerges from interactions within the network structure. Here we study one-dimensional (1D) spatial structures and focus on sparse non-Hermitian random matrices in the spirit of tight-binding models in solid state physics.

Note: Fast compact laser shutter using a direct current motor and three-dimensional printing.

We present a mechanical laser shutter design that utilizes a direct current electric motor to rotate a blade which blocks and unblocks a light beam. The blade and the main body of the shutter are modeled with computer aided design (CAD) and are produced by 3D printing. Rubber flaps are used to limit the blade's range of motion, reducing vibrations and preventing undesirable blade oscillations.