Waves of maximal height for a class of nonlocal equations with homogeneous symbols

We discuss the existence and regularity of periodic traveling-wave solutions of a class of nonlocal equations with homogeneous symbol of order $-r$, where $r>1$. Based on the properties of the nonlocal convolution operator, we apply analytic bifurcation theory and show that a highest, peaked, periodic traveling-wave solution is reached as the limiting case at the end of the main bifurcation curve. The regularity of the highest wave is proved to be exactly Lipschitz. As an application of our analysis, we reformulate the steady reduced Ostrovsky equation in a nonlocal form in terms of a Fourier multiplier operator with symbol $m(k)=k^{-2}$. Thereby we recover its unique highest $2\pi$-periodic, peaked traveling-wave solution, having the property of being exactly Lipschitz at the crest.Comment: 25 page

P-d scattering and the bound state.

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Maximal quantum randomness in Bell tests

The non-local correlations exhibited when measuring entangled particles can be used to certify the presence of genuine randomness in Bell experiments. While non-locality is necessary for randomness certification, it is unclear when and why non-locality certifies maximal randomness. We provide here a simple argument to certify the presence of maximal local and global randomness based on symmetries of a Bell inequality and the existence of a unique quantum probability distribution that maximally violates it. Using our findings, we prove the existence of N-party Bell test attaining maximal global randomness, that is, where a combination of measurements by each party provides N perfect random bits.Comment: 5 pages, 1 figur