New Sign Uncertainty Principles

We prove new sign uncertainty principles which vastly generalize the recent developments of Bourgain, Clozel & Kahane and Cohn & Gon\c{c}alves, and apply our results to a variety of spaces and operators. In particular, we establish new sign uncertainty principles for Fourier and Dini series, the Hilbert transform, the discrete Fourier and Hankel transforms, spherical harmonics, and Jacobi polynomials, among others. We present numerical evidence highlighting the relationship between the discrete and continuous sign uncertainty principles for the Fourier and Hankel transforms, which in turn are connected with the sphere packing problem via linear programming. Finally, we explore some connections between the sign uncertainty principle on the sphere and spherical designs.Comment: 45 pages, 2 figures, 3 tables, v3: typos corrected, numerics extende

Bell nonlocality

Bell's 1964 theorem, which states that the predictions of quantum theory cannot be accounted for by any local theory, represents one of the most profound developments in the foundations of physics. In the last two decades, Bell's theorem has been a central theme of research from a variety of perspectives, mainly motivated by quantum information science, where the nonlocality of quantum theory underpins many of the advantages afforded by a quantum processing of information. The focus of this review is to a large extent oriented by these later developments. We review the main concepts and tools which have been developed to describe and study the nonlocality of quantum theory, and which have raised this topic to the status of a full sub-field of quantum information science.Comment: 65 pages, 7 figures. Final versio

Computer-assisted Existence Proofs for Navier-Stokes Equations on an Unbounded Strip with Obstacle

The incompressible stationary 2D Navier-Stokes equations are considered on an unbounded strip domain with a compact obstacle. First, a computer-assisted existence and enclosure result for the velocity (in a suitable divergence-free Sobolev space) is presented. Starting from an approximate solution (computed with divergence-free finite elements), we determine a bound for its defect, and a norm bound for the inverse of the linearization at the approximate solution. For the latter, bounds for the essential spectrum and for eigenvalues play a crucial role, especially for the eigenvalues ``close to\u27\u27 zero. Note that, on an unbounded domain, the only general method for computing the desired norm bound appears to be via eigenvalue bounds. To obtain the desired lower bounds for the eigenvalues below the essential spectrum we use the Rayleigh-Ritz method, a corollary of the Temple-Lehmann theorem and a homotopy method. Finally, if the computer-assisted proof provides the existence of a velocity field, the existence of a corresponding pressure can be obtained by purely analytical techniques. Nevertheless, for a given approximate solution to the pressure our methods provide an error bound (in a dual norm) as well