Tropical Markov dynamics and Cayley cubic

We study the tropical version of Markov dynamics on the Cayley cubic, introduced by V.E. Adler and one of the authors. We show that this action is semi-conjugated to the standard action of $SL_2(\mathbb Z)$ on a torus, and thus is ergodic with the Lyapunov exponent and entropy given by the logarithm of the spectral radius of the corresponding matrix.Comment: Extended version, accepted for publication in "Integrable Systems and Algebraic Geometry" (Editors: R. Donagi, T. Shaska), Cambridge Univ. Press: LMS Lecture Notes Series, 201

Critical Entanglement for the Half-Filled Extended Hubbard Model

We study the ground state of the one-dimensional extended Hubbard model at half-filling using the entanglement entropy calculated by Density Matrix Renormalization Group (DMRG) techniques. We apply a novel curve fitting and scaling method to accurately identify a $2^{nd}$ order critical point as well as a Berezinskii-Kosterlitz-Thouless (BKT) critical point. Using open boundary conditions and medium-sized lattices with very small truncation errors, we are able to achieve similar accuracy to previous authors. We also report observations of finite-size and boundary effects that can be remedied with careful pinning.Comment: 10 pages, 12 figure

Prediction of hydrodynamics and chemistry of confined turbulent methane-air flames with attention to formation of oxides of nitrogen

A formulation of the governing partial differential equations for fluid flow and reacting chemical species in a tubular combustor is presented. A numerical procedure for the solution of the governing differential equations is described, and models for chemical equilibrium and chemical kinetics calculations are presented. The chemical equilibrium model is used to characterize the hydrocarbon reactions. The chemical kinetics model is used to predict the concentrations of the oxides of nitrogen. The combustor consists of a cylindrical duct of varying cross sections with concentric streams of gaseous fuel and air entering the duct at one end. Four sample cases with specified inlet and boundary conditions are considered, and the results are discusse

Conway river and Arnold sail

We establish a simple relation between two geometric constructions in number theory: the Conway river of a real indefinite binary quadratic form and the Arnold sail of the corresponding pair of lines.Comment: Slightly modified version to appear in Arnold Mathematical Journa

Optical Modelling and Phylogenetic Analysis Provide Clues to the Likely Function of Corneal Nipple Arrays in Butterflies and Moths.

This is the final version. Available from MDPI via the DOI in this recordThe lenses in compound eyes of butterflies and moths contain an array of nipple-shaped protuberances, or corneal nipples. Previous work has suggested that these nipples increase light transmittance and reduce the eye glare of moths that are inactive during the day. This work builds on but goes further than earlier analyses suggesting a functional role for these structures including, for the first time, an explanation of why moths are attracted to UV light. Using a phylogenetic approach and 3D optical modelling, we show empirically that these arrays have been independently lost from different groups of moths and butterflies and vary within families. We find differences in the shape of nipples between nocturnal and diurnal species, and that anti-glow reflectance levels are different at different wave-lengths, a result thereby contradicting the currently accepted theory of eye glow for predator avoidance. We find that there is reduced reflectance, and hence greater photon absorption, at UV light, which is probably a reason why moths are attracted to UV. We note that the effective refractive index at the end of the nipples is very close to the refractive index of water, allowing almost all the species with nipples to see without distortion when the eye is partially or completely wet and providing the potential to keep eyes dry. These observations provide a functional explanation for these arrays. Of special interest is the finding that their repeated and independent loss across lepidopteran phylogeny is inconsistent with the explanation that they are being lost in the 'higher', more active butterflies

Prediction of hydrodynamics and chemistry of confined turbulent methane-air frames in a two concentric tube combustor

A formulation of the governing partial differential equations for fluid flow and reacting chemical species in a two-concentric-tube combustor is presented. A numerical procedure for the solution of the governing differential equations is described and models for chemical-equilibrium and chemical-kinetics calculations are presented. The chemical-equilibrium model is used to characterize the hydrocarbon reactions. The chemical-kinetics model is used to predict the concentrations of the oxides of nitrogen. The combustor considered consists of two coaxial ducts. Concentric streams of gaseous fuel and air enter the inlet duct at one end; the flow then reverses and flows out through the outer duct. Two sample cases with specified inlet and boundary conditions are considered and the results are discussed

Fast localized wavefront correction using area-mapped phase-shift interferometry

We propose an innovative method for localized wavefront correction based on area-mapped phase-shift (AMPS) interferometry. In this Letter, we present the theory and then experimentally compare it with a previously demonstrated method based on spot-optimized phase-stepping (SOPS) interferometry. We found that AMPS outperforms SOPS interferometry in terms of speed by threefold, although in noisy environments the improvements may be larger. AMPS yielded similar point-spread functions (PSF) as SOPS for moderate system-induced aberrations, but yielded a slightly less ideal PSF for larger aberrations. The method described in this Letter may prove crucial for applications where the phase-stepping solution does not have sufficient speed