Arithmetic properties of the Ramanujan function

We study some arithmetic properties of the Ramanujan function $\tau(n)$, such as the largest prime divisor $P(\tau(n))$ and the number of distinct prime divisors $\omega(\tau(n))$ of $\tau(n)$ for various sequences of $n$. In particular, we show that \hbox{$P(\tau(n)) \geq (\log n)^{33/31 + o(1)}$} for infinitely many $n$, and \begin{equation*} P(\tau(p)\tau(p^2)\tau(p^3)) > (1+o(1))\frac{\log\log p\log\log\log p} {\log\log\log\log p} \end{equation*} for every prime $p$ with \hbox{$\tau(p)\neq 0$}.Comment: 8 page

Trihamiltonian extensions of separable systems in the plane

A method to construct trihamiltonian extensions of a separable system is presented. The procedure is tested for systems, with a natural Hamiltonian, separable in classical sense in one of the four orthogonal separable coordinate systems of the Euclidean plane, and some explicit examples are constructed. Finally a conjecture on possible generalizations to other classes of systems is discussed: in particular, the method can be easily adapted to the eleven orthogonal separable coordinate sets of the Euclidean three-space.Comment: 20 page

Acoustical Absorption and Scattering Cross-Sections of Spherical Bubble Clouds

The present work investigates the acoustical absorption and scattering cross-sections of spherical bubble clouds subject to harmonic far field pressure excitation. Bubble dynamics effects and energy dissipation due to viscosity, heat transfer, liquid compressibility and relative motion of the two phases are included. The equations of motion for the average flow and for the bubble radius are linearized and a closed form solution is obtained. Due to the presence of natural oscillatory modes and frequencies, the acoustical cross-sections of the cloud are very different from those of each individual bubble in the cloud, as well as from the acoustical cross-sections of a single large bubble with the same volume of vapor and gas. In general the acoustical properties of any given volume of the dispersed phase depend strongly on the degree of dispersion because of the complex interactions of the dynamics of the bubbles with the whole flow