Effective speed of sound in phononic crystals

A new formula for the effective quasistatic speed of sound $c$ in 2D and 3D periodic materials is reported. The approach uses a monodromy-matrix operator to enable direct integration in one of the coordinates and exponentially fast convergence in others. As a result, the solution for $c$ has a more closed form than previous formulas. It significantly improves the efficiency and accuracy of evaluating $c$ for high-contrast composites as demonstrated by a 2D example with extreme behavior.Comment: 4 pages, 1 figur

Hall-MHD small-scale dynamos

Much of the progress in our understanding of dynamo mechanisms has been made within the theoretical framework of magnetohydrodynamics (MHD). However, for sufficiently diffuse media, the Hall effect eventually becomes non-negligible. We present results from three dimensional simulations of the Hall-MHD equations subjected to random non-helical forcing. We study the role of the Hall effect in the dynamo efficiency for different values of the Hall parameter, using a pseudospectral code to achieve exponentially fast convergence. We also study energy transfer rates among spatial scales to determine the relative importance of the various nonlinear effects in the dynamo process and in the energy cascade. The Hall effect produces a reduction of the direct energy cascade at scales larger than the Hall scale, and therefore leads to smaller energy dissipation rates. Finally, we present results stemming from simulations at large magnetic Prandtl numbers, which is the relevant regime in hot and diffuse media such a the interstellar medium.Comment: 11 pages and 11 figure

Ergodicity of conservative communication networks

Projet MEVALWe analyze a communication network with several types of calls. For a wide class of conservative service disciplines, we give ergodicity criteria. Exponentially fast convergence to steady state is also proved

How many digits are needed?

Let $X_1,X_2,...$ be the digits in the base-$q$ expansion of a random variable $X$ defined on $[0,1)$ where $q\ge2$ is an integer. For $n=1,2,...$, we study the probability distribution $P_n$ of the (scaled) remainder $\sum_{k=n+1}^\infty X_qq^{n-k}$: If $X$ has an absolutely continuous CDF then $P_n$ converges in the total variation metric to Lebesgue measure on the unit interval; under certain smoothness conditions we establish exponentially fast convergence of $P_n$ and its PDF $f_n$; and we give examples of these results. The results are extended to the case of a multivariate random variable defined on a unit cube.Comment: 15 pages, 2 figure