TRANSLATIONAL-ROTATIONAL MOTION OF A NONSTATIONARY AXISYMMETRIC BODY

A nonstationary two-body problem is considered such that one of the bodies has a spherically symmetric density distribution and is central, while the other one is a satellite with axisymmetric dynamical structure, shape, and variable oblateness. Newton’s interaction force is characterized by an approximate expression of the force function up to the second harmonic. The masses of the central body and the satellite vary isotropically at different rates and do not occur reactive forces and additional rotational moments. The nonstationary axisymmetric body have an equatorial plane of symmetry. Thus, it has three mutually perpendicular planes of symmetry. The axes of its intrinsic coordinate system coincide with the principal axes of inertia and they are directed along the intersection lines of these three mutually perpendicular planes. This position remains unchangeable during the evolution. Equations of motion of the satellite in a relative system of coordinates are considered. The translational- rotational motion of the nonstationary axisymmetric body in the gravitational field of the nonstationary ball is studied by perturbation theory methods. The equations of secular perturbations reduces to the fourth order system with one first integral. This first integral is considered and three-dimensional graphs of this first integral are plotted using the Wolfram Mathematica system.

Mixing induced by ISWs breaking over a sloping boundary: an analytical heuristic model

<p>We propose an analytical approach to estimate mixing efficiency in Internal Solitary Waves (ISWs) breaking processes. We make use of the theoretical framework of Winters et al. [1995] to describe the energetics of a stratified fluid flow, calculating the Available Potential Energy (APE) of an ISW of depression in a two-layer system, assuming that the symmetric density structure on both sides of the feature is exactly the same. Starting from the definition of mixing efficiency given by Michallet and Ivey [1999], through the Ozmidov and Thorpe length-scales we derive an expression for the mixing efficiency avoiding the use of any wave model (as KdV-type models or strongly nonlinear models) to estimate the wave energy. The model is successfully verified through laboratory experiments performed in a wave tank and is meant to be applied by using real field CTD casts.</p><p> </p><p>References:</p><p>Winters, K., Lombard, P., Riley, J., and D’Asaro, E. (1995). <em>Available potential</em></p><p><em>energy and mixing in density-stratified fluids</em>. J. Fluid Mech., 289, 115-128.</p><p>Michallet, H. and Ivey, G. (1999). <em>Experiments on mixing due to internal solitary</em></p><p><em>waves breaking on uniform slopes</em>. Journal of Geophysical Research: Oceans,</p><p>104(C6), 13467-13477</p>

Problems of Signal Processing in Ultrasonic Gas Flow Measurement

Vortex measuring methods with ultrasound are distinguished by small bluff bodies, low pressure losses and high sensitivity. The ultrasound wave is modulated by the vortices behind the bluff body. The modulation frequency represents the flow velocity and can be determined by well-known demodulation procedures.Cross correlation methods use the natural turbulences in a fluid. Because of the skewed density function of the velocity components the maximum of the cross correlation function does not represent the transit time of the turbulences between two ultrasonic barriers. Processing of the complex modulated signal is very difficult because the phase of the signal can reach very high values and can not be considered unambiguously. It is advantageous to simplify the signal processing by artificially generated vortices by a small bluff body. It results in a symmetric density distribution and symmetric cross correlation function. Furthermore, it results in a self-monitoring system. Alternatively, two different carrier frequencies can be applied to the two ultrasonic waves. In the cross correlation function the carrier frequencies are eliminated automatically.

Estimating the division kernel of a size-structured population

We consider a size-structured model describing a population of cells proliferating by division. Each cell contain a quantity of toxicity which grows linearly according to a constant growth rate α. At division, the cells divide at a constant rate R and share their content between the two daughter cells into fractions Γ and 1 − Γ where Γ has a symmetric density h on [ 0,1 ], since the daughter cells are exchangeable. We describe the cell population by a random measure and observe the cells on the time interval [ 0,T ] with fixed T. We address here the problem of estimating the division kernel h (or fragmentation kernel) when the division tree is completely observed. An adaptive estimator of h is constructed based on a kernel function K with a fully data-driven bandwidth selection method. We obtain an oracle inequality and an exponential convergence rate, for which optimality is considered.

METHODS FOR SYMMETRIZING RANDOM VARIABLES

Let X be a random variable with nonsymmetric density p(x). We give the symmetric density q(x) closest to it in the sense of Kulback–Liebler and Hellinger distances. (All symmetries are around zero.) For the first distance, we show that q(x) is proportional to the geometric mean of p(x) and p(−x). For example, a symmetrized shifted exponential is a centered uniform, and a symmetrized shifted gamma is a centered beta random variable. For the second distance, q(x) is proportional to the square of the arithmetic mean of p(x)1/2 and p(−x)1/2. Sample versions are also given for each. We also give the optimal random function f such that f(X) is symmetrically distributed and minimizes |f(X)−X|. Finally, we show how to optimize the Hellinger distance for vector X subject to supersymmetry and for scalar X subject to being monotone about zero in each half-line.