Gemunu H. Gunaratne

Scaling of hard thermal turbulence in Rayleigh-Bénard convection

An experimental study of Rayleigh-Benard convection in helium gas at roughly 5 K is performed in a cell with aspect ratio 1. Data are analysed in a ‘hard turbulence’ region (4 × 10 7 Ra 12 ) in which the Prandtl number remains between 0.65 and 1.5. The main observation is a simple scaling behaviour over this entire range of Ra . However the results are not the same as in previous theories. For example, a classical result gives the dimensionless heat flux, Nu , proportional to

Markov Processes, Hurst Exponents, and Nonlinear Diffusion Equations with application to finance

Ra^{\frac{1}{3}}

Rhombic patterns: Broken hexagonal symmetry

while experiment gives an index much closer to

Enhanced quantum interference effects in parallel Josephson junction arrays

\frac{2}{7}

Open‐field arena boundary is a primary object of exploration for Drosophila

. A new scaling theory is described. This new approach suggests scaling indices very close to the observed ones. The new approach is based upon the assumption that the boundary layer remains in existence even though its Rayleigh number is considerably greater than unity and is, in fact, diverging. A stability analysis of the boundary layer is performed which indicates that the boundary layer may be stabilized by the interaction of buoyancy driven effects and a fluctuating wind.

Martingale Option Pricing

We show by explicit closed form calculations that a Hurst exponent H≠12 does not necessarily imply long time correlations like those found in fractional Brownian motion (fBm). We construct a large set of scaling solutions of Fokker–Planck partial differential equations (pdes) where H≠12. Thus Markov processes, which by construction have no long time correlations, can have H≠12. If a Markov process scales with Hurst exponent H≠12 then it simply means that the process has nonstationary increments. For the scaling solutions, we show how to reduce the calculation of the probability density to a single integration once the diffusion coefficient D(x,t) is specified. As an example, we generate a class of student-t-like densities from the class of quadratic diffusion coefficients. Notably, the Tsallis density is one member of that large class. The Tsallis density is usually thought to result from a nonlinear diffusion equation, but instead we explicitly show that it follows from a Markov process generated by a linear Fokker–Planck equation, and therefore from a corresponding Langevin equation. Having a Tsallis density with H≠12 therefore does not imply dynamics with correlated signals, e.g., like those of fBm. A short review of the requirements for fBm is given for clarity, and we explain why the usual simple argument that H≠12 implies correlations fails for Markov processes with scaling solutions. Finally, we discuss the question of scaling of the full Green function g(x,t;x′,t′) of the Fokker–Planck pdes.

Martingales, detrending data, and the efficient market hypothesis

Landau-Ginzburg equations derived to conserve two-dimensional spatial symmetries lead to the prediction that rhombic arrays with characteristic angles slightly differ from 60 degrees should form in many systems. Beyond the bifurcation from the uniform state to patterns, rhombic patterns are linearly stable for a band of angles near the 60 degrees angle of regular hexagons. Experiments conducted on a reaction-diffusion system involving a chlorite-iodide-malonic acid reaction yield rhombic patterns in good accord with the theory.

Universal strange attractors on wrinkled tori

We have calculated the total critical current as a function of applied magnetic flux for a superconducting interferometer consisting of many Josephson junctions in parallel. An enhancement and narrowing of the periodic principal maxima in the critical current versus flux characteristic is predicted as the number of junctions in parallel increases, even when the effects of finite self‐ and mutual inductances and nonuniformity of the junction critical currents are included in the calculations. The possible application of the superconducting quantum interference grating, or SQUIG, for the detection of magnetic flux is discussed.

ASYMMETRIC CELLS AND ROTATING RINGS IN CELLULAR FLAMES

Drosophila adults, when placed into a novel open‐field arena, initially exhibit an elevated level of activity followed by a reduced stable level of spontaneous activity and spend a majority of time near the arena edge, executing motions along the walls. In order to determine the environmental features that are responsible for the initial high activity and wall‐following behavior exhibited during exploration, we examined wild‐type and visually impaired mutants in arenas with different vertical surfaces. These experiments support the conclusion that the wall‐following behavior of Drosophila is best characterized by a preference for the arena boundary, and not thigmotaxis or centrophobicity. In circular arenas, Drosophila mostly move in trajectories with low turn angles. Since the boundary preference could derive from highly linear trajectories, we further developed a simulation program to model the effects of turn angle on the boundary preference. In an hourglass‐shaped arena with convex‐angled walls that forced a straight versus wall‐following choice, the simulation with constrained turn angles predicted general movement across a central gap, whereas Drosophila tend to follow the wall. Hence, low turn angled movement does not drive the boundary preference. Lastly, visually impaired Drosophila demonstrate a defect in attenuation of the elevated initial activity. Interestingly, the visually impaired w1118 activity decay defect can be rescued by increasing the contrast of the arenas edge, suggesting that the activity decay relies on visual detection of the boundary. The arena boundary is, therefore, a primary object of exploration for Drosophila.

Modeling Drosophila Positional Preferences in Open Field Arenas with Directional Persistence and Wall Attraction

We show that our earlier generalization of the Black–Scholes partial differential equation (pde) for variable diffusion coefficients is equivalent to a Martingale in the risk neutral discounted stock price. Previously, the equivalence of Black–Scholes to a Martingale was proven for the case of the Gaussian returns model by Harrison and Kreps, but we prove it for a much larger class of returns models where the returns diffusion coefficient depends irreducibly on both returns x and time t. That option prices blow up if fat tails in logarithmic returns x are included in market return is also proven.