Existence of Gaussian cubature formulas

We provide a necessary and sufficient condition for existence of Gaussian cubature formulas. It consists of checking whether some overdetermined linear system has a solution and so complements Mysovskikh's theorem which requires computing common zeros of orthonormal polynomials. Moreover, the size of the linear system shows that existence of a cubature formula imposes severe restrictions on the associated linear functional. For fixed precision (or degree), the larger the number of variables the worse it gets. And for fixed number of variables, the larger the precision the worse it gets. Finally, we also provide an interpretation of the necessary and sufficient condition in terms of existence of a polynomial with very specific properties

Bounding the support of a measure from its marginal moments

Given all moments of the marginals of a measure on Rn, one provides (a) explicit bounds on its support and (b), a numerical scheme to compute the smallest box that contains the support. It consists of solving a hierarchy of generalized eigenvalue problems associated with some Hankel matrices.Comment: To appear in Proc. Amer. Math. So

Light Sterile Neutrinos in Particle Physics: Experimental Status

Most of the neutrino oscillation results can be explained by the three-neutrino paradigm. However several anomalies in short baseline oscillation data could be interpreted by invoking a hypothetical fourth neutrino, separated from the three standard neutrinos by a squared mass difference of more than 0.1 eV$^2$. This new neutrino, often called sterile, would not feel standard model interactions but mix with the others. Such a scenario calling for new physics beyond the standard model has to be either ruled out or confirmed with new data. After a brief review of the anomalous oscillation results we discuss the world-wide experimental proposal aiming to clarify the situation.Comment: 14 pages, 2 figures. To appear in the proceedings of the 13th International Conference on Topics in Astroparticle and Underground Physics, TAUP 2013 (F. Avignone & W. Haxton, editors, Physics Procedia, Elsevier) ; Minor revisions in version

Recovering an homogeneous polynomial from moments of its level set

Let $K:={x: g(x)\leq 1}$ be the compact sub-level set of some homogeneous polynomial $g$. Assume that the only knowledge about $K$ is the degree of $g$ as well as the moments of the Lebesgue measure on $K$ up to order 2d. Then the vector of coefficients of $g$ is solution of a simple linear system whose associated matrix is nonsingular. In other words, the moments up to order 2d of the Lebesgue measure on $K$ encode all information on the homogeneous polynomial $g$ that defines $K$ (in fact, only moments of order $d$ and 2d are needed)

A new look at nonnegativity on closed sets and polynomial optimization

We first show that a continuous function f is nonnegative on a closed set $K\subseteq R^n$ if and only if (countably many) moment matrices of some signed measure $d\nu =fd\mu$ with support equal to K, are all positive semidefinite (if $K$ is compact $\mu$ is an arbitrary finite Borel measure with support equal to K. In particular, we obtain a convergent explicit hierarchy of semidefinite (outer) approximations with {\it no} lifting, of the cone of nonnegative polynomials of degree at most $d$. Wen used in polynomial optimization on certain simple closed sets \K (like e.g., the whole space $\R^n$, the positive orthant, a box, a simplex, or the vertices of the hypercube), it provides a nonincreasing sequence of upper bounds which converges to the global minimum by solving a hierarchy of semidefinite programs with only one variable. This convergent sequence of upper bounds complements the convergent sequence of lower bounds obtained by solving a hierarchy of semidefinite relaxations