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

Mechanical Evidence of the Orbital Angular Momentum to Energy Ratio of Vortex Beams

We measure, in a single experiment, both the radiation pressure and the torque due to a wide variety of propagating acoustic vortex beams. The results validate, for the first time directly, the theoretically predicted ratio of the orbital angular momentum to linear momentum in a propagating beam. We experimentally determine this ratio using simultaneous measurements of both the levitation force and the torque on an acoustic absorber exerted by a broad range of helical ultrasonic beams produced by a 1000-element matrix transducer array. In general, beams with helical phase fronts have been shown to contain orbital angular momentum as the result of the azimuthal component of the Poynting vector around the propagation axis. Theory predicts that for both optical and acoustic helical beams the ratio of the angular momentum current of the beam to the power should be given by the ratio of the beam’s topological charge to its angular frequency. This direct experimental observation that the ratio of the torque to power does convincingly match the expected value (given by the topological charge to angular frequency ratio of the beam) is a fundamental result

Combustion of hydrogen-air jets in local chemical equilibrium: A guide to the CHARNAL computer program

A guide to a computer program, written in FORTRAN 4, for predicting the flow properties of turbulent mixing with combustion of a circular jet of hydrogen into a co-flowing stream of air is presented. The program, which is based upon the Imperial College group's PASSA series, solves differential equations for diffusion and dissipation of turbulent kinetic energy and also of the R.M.S. fluctuation of hydrogen concentration. The effective turbulent viscosity for use in the shear stress equation is computed. Chemical equilibrium is assumed throughout the flow

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

Twisted [(R3P)PdX] groups above dicarbaborane ligands: 4-dimethylsulfido-3-iodo-3-triphenylphosphine-closo-3-pallada-1,2-dicarbadodecaborane and 3-dimethylphenylphosphine-3-chloro-4-dimethylsulfido-closo-3-pallada-1,2-dicarbadodecaborane

The structural analyses of [3-(PPh₃)-3-I-4-(SMe₂)-closo-3,1,2-PdC₂B₉H₁₀] or [Pd(C₄H₁₆B₉S)I(C₁₈H₁₅P)], (I), and [3-(PPhMe₂)-3-Cl-4-(SMe₂)-closo-3,1,2-PdC₂B₉H₁₀] or [Pd(C₄H₁₆B₉S)Cl(C₈H₁₁P)], (II), show that in comparison with [3-(PR₃)2-closo-3,1,2-PdC₂B₉H₁₁] the presence of the 4-SMe₂ group causes the [PdX(PR₃)] unit (X = halogen) to twist about an axis passing through the Pd atom and the directly opposite B atom of the carbaborane ligand. The halogen atoms are located almost directly above a C atom in the C₂B₃ face, and the conformations of the [PdX(PR₃)] units above the C₂B₃ faces are not those predicted from molecular orbital calculations of the closo-3,1,2-PdC₂B₉ system. The fact that the variation from the predicted conformation is greater in the case of (I) than in (II) may be ascribed to the greater steric interactions induced by the I atom in (I) compared with the Cl atom in (II)