The existence of two inertial ranges forced homogeneous two-dimensional turbulence was conjectured about half a century ago. Given forcing, confined to typical wavenumber ${k}_{f}$, direct cascade with ${k}^{\ensuremath{-}3}$ scaling for $k\ensuremath{\gg}{k}_{f}$ and an inverse ${k}^{\ensuremath{-}5/3}$ $k\ensuremath{\ll}{k}_{f}$ would be established. Using geometric integrator on unit sphere ...

The Fast Fourier Transform was applied to spatial spectrum modeling of a one-dimensional fractal (Cantor set), a two-dimensional fractal (Sierpinski carpet), and a three-dimensional fractal (Menger sponge). A spectrum is developed for different levels. The spatial spectrum was also obtained and modeled for various filling parameters. The ParaView software package was used for 3D modeling.

Four-dimensional NMR spectroscopy can be speeded up by a large factor by a projection-reconstruction technique related to that used in X-ray tomography. The limited amount of information recorded in a few suitably-tilted projection planes suffices to recreate the entire four-dimensional spectrum. The method is demonstrated by reference to the 500 MHz spectrum of 1 millimolar aqueous solution of...