<p style='text-indent:20px;'>We prove global in time dispersion for the wave and Klein-Gordon equation inside Friedlander domain by taking full advantage of space-time localization caustics a precise estimate number waves that may cross at given, large time. Moreover, we uncover significant difference between low frequency, regime, where exhibits worse decay than wave, unlike flat space.&...

We study experimentally blast wave dynamics on a weakly interacting fluid of light. The density and velocity are measured in 1D 2D geometries. Using state equation arising from the analogy between optical propagation paraxial approximation hydrodynamic Euler's equation, we access hydrostatic dynamic pressure. In configuration, observe negative differential pressure after fast expansion localize...

Given two relatively prime positive integers m < n we consider the smallest positive solution (x0, y0) to the equation mx−ny = 1. E. I. Dinaburg and Y. G. Sinai have used continued fractions to show that the ratios x0/n are uniformly distributed in [0, 1], when n and m run through consequtive integers of intervals of comparable sizes. We use a bound of exponential sums due to W. Duke, J. B. Fri...

Conjecture 32.1 is easily seen to be true for a torus (i.e., G G r m for some r 0), but even the simplest non-trivial case (that of G SL2 ) remains inaccessible. Guido and I published 5 papers together, all in some sense connected with this conjecture. We used the integral form GZ of a complex reductive algebraic group (which is a group scheme over Spec Z ) in order to form the group G(F) of po...