Abstract:
It is now widely accepted that a new middle baseline disappearance reactor neutrino experiment with multiple detectors could provide a clean measurement of the $\theta_{13}$ mixing angle, free from any parameter degeneracies and correlations induced by matter effect and the unknown leptonic Dirac CP phase. The current best constraint on the third mixing angle comes from the Chooz reactor neutrino experiment $\sin^{2}(2\theta_{13})<0.2$ (90$ %$ C.L., $\Delta m_{\rm atm}^{2}=2.0 10^{-3}$ eV$^{2}$). Several projects of experiment, with different timescales, have been proposed over the last two years all around the world. Their sensitivities range from $\sin^{2}(2\theta_{13})<$ 0.01 to 0.03, having thus an excellent discovery potential of the $\nu_e$ fraction of $\nu_3$.

Abstract:
The Double-Chooz experiment goal is to search for a non-vanishing value of the Theta13 neutrino mixing angle. This is the last step to accomplish prior moving towards a new era of precision measurements in the lepton sector. The current best constraint on the third mixing angle comes from the CHOOZ reactor neutrino experiment $\sin(2\theta_{13})^{2}<0.2$ (90% C.L., $\Delta m_{atm}^{2}=2.0$ eV$^{2}$). Double-Chooz will explore the range of $\sin(2\theta_{13})^{2}$ from 0.2 to 0.03-0.02, within three years of data taking. The improvement of the CHOOZ result requires an increase in the statistics, a reduction of the systematic error below one percent, and a careful control of the backgrounds. Therefore, Double-Chooz will use two identical detectors, one at 150 m and another at 1.05 km distance from the Chooz nuclear cores. In addition, we will to use the near detector as a ``state of the art'' prototype to investigate the potential of neutrinos for monitoring the civil nuclear power plants. The plan is to start operation with two detectors in 2008, and to reach a sensitivity sin$^{2}$$(2\theta_{13})$ of 0.05 in 2009, and 0.03-0.02 in 2011.

Abstract:
Several observed anomalies in neutrino oscillation data could be explained by a hypothetical fourth neutrino separated from the three standard neutrinos by a squared mass difference of a few 0.1 eV$^2$ or more. This hypothesis can be tested with MCi neutrino electron capture sources ($^{51}$Cr) or kCi antineutrino $\beta$-source ($^{144}$Ce) deployed inside or next to a large low background neutrino detector. In particular, the compact size of this source coupled with the localization of the interaction vertex lead to an oscillating pattern in event spatial (and possibly energy) distributions that would unambiguously determine neutrino mass differences and mixing angles.

Abstract:
Recently new reactor antineutrino spectra have been provided for 235U, 239Pu, 241Pu and 238U, increasing the mean flux by about 3 percent. To good approximation, this reevaluation applies to all reactor neutrino experiments. The synthesis of published experiments at reactor-detector distances <100 m leads to a ratio of observed event rate to predicted rate of 0.976(0.024). With our new flux evaluation, this ratio shifts to 0.943(0.023), leading to a deviation from unity at 98.6% C.L. which we call the reactor antineutrino anomaly. The compatibility of our results with the existence of a fourth non-standard neutrino state driving neutrino oscillations at short distances is discussed. The combined analysis of reactor data, gallium solar neutrino calibration experiments, and MiniBooNE-neutrino data disfavors the no-oscillation hypothesis at 99.8% C.L. The oscillation parameters are such that |Delta m_{new}^2|>1.5 eV^2 (95%) and sin^2(2\theta_{new})=0.14(0.08) (95%). Constraints on the theta13 neutrino mixing angle are revised.

Abstract:
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.

Abstract:
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.

Abstract:
Given a nonnegative polynomial f, we provide an explicit expression for its best $\ell_1$-norm approximation by a sum of squares of given degree.

Abstract:
We consider the semi-infinite optimization problem: $f^*:=\min_{x\in X}\:\{f(x): g(x,y)\,\leq \,0,\:\forally\in Y_x\}$, where $f,g$ are polynomials and $X\subset R^n$ as well as $Y_\x\subset R^p$, $x\in X$, are compact basic semi-algebraic sets. To approximate $f^*$ we proceed in two steps. First, we use the "joint+marginal" approach of the author to approximate from above the function $x\mapsto\Phi(x)=\sup \{g(x,y): y\in Y_x\}$ by a polynomial $\Phi_d\geq\Phi$, of degree at most $2d$, with the strong property that $\Phi_d$ converges to $\Phi$ for the $L_1$-norm, as $d\to\infty$ (and in particular, almost uniformly for some subsequence $(d_\ell)$, $\ell\in\N$). Then we solve the polynomial optimization problem $f^*_d=\min_{x\in X} \{f(x): \Phi_d(x)\leq0\}$ via a (by now standard) hierarchy of semidefinite relaxations. It turns out that the optimal value $f^*_d\geq f^*$ converges to $f^*$ as $d\to\infty$. In practice we let $d$ be fixed, small, and relax the constraint $\Phi_d\leq0$ to $\Phi_d(x)\leq\epsilon$ with $\epsilon>0$, allowing to change $\epsilon$ dynamically.

Abstract:
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).

Abstract:
We consider the general polynomial optimization problem $P: f^*=\min \{f(x)\,:\,x\in K\}$ where $K$ is a compact basic semi-algebraic set. We first show that the standard Lagrangian relaxation yields a lower bound as close as desired to the global optimum $f^*$, provided that it is applied to a problem $\tilde{P}$ equivalent to $P$, in which sufficiently many redundant constraints (products of the initial ones) are added to the initial description of $P$. Next we show that the standard hierarchy of LP-relaxations of $P$ (in the spirit of Sherali-Adams' RLT) can be interpreted as a brute force simplification of the above Lagrangian relaxation in which a nonnegative polynomial (with coefficients to be determined) is replaced with a constant polynomial equal to zero. Inspired by this interpretation, we provide a systematic improvement of the LP-hierarchy by doing a much less brutal simplification which results into a parametrized hierarchy of semidefinite programs (and not linear programs any more). For each semidefinite program in the parametrized hierarchy, the semidefinite constraint has a fixed size $O(n^k)$, independently of the rank in the hierarchy, in contrast with the standard hierarchy of semidefinite relaxations. The parameter $k$ is to be decided by the user. When applied to a non trivial class of convex problems, the first relaxation of the parametrized hierarchy is exact, in contrast with the LP-hierarchy where convergence cannot be finite. When applied to 0/1 programs it is at least as good as the first one in the hierarchy of semidefinite relaxations. However obstructions to exactness still exist and are briefly analyzed. Finally, the standard semidefinite hierarchy can also be viewed as a simplification of an extended Lagrangian relaxation, but different in spirit as sums of squares (and not scalars) multipliers are allowed.