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Spatial or geometric quantiles are among the most celebrated concepts of multivariate quantiles. The spatial quantile \(\mu _{\alpha ,u}(P)\) of a probability measure P over \(\mathbb {R}^d\) is a point in \(\mathbb R^d\) indexed by an order \(\alpha \in [0,1)\) and a direction u in the unit sphere \(\mathcal {S}^{d-1}\) of \(\mathbb R^d\)—or equivalently by a vector \(\alpha u\) in the open unit ball of \(\mathbb R^d\). Recently, Girard and Stupfler (2017) proved that (i) the extreme quantiles \(\mu _{\alpha ,u}(P)\) obtained as \(\alpha \rightarrow 1\) exit all compact sets of \(\mathbb R^d\) and that (ii) they do so in a direction converging to u. These results help understanding the nature of these quantiles: the first result is particularly striking as it holds even if P has a bounded support, whereas the second one clarifies the delicate dependence of spatial quantiles on u. However, they were established under assumptions imposing that P is non-atomic, so that it is unclear whether they hold for empirical probability measures. We improve on this by proving these results under much milder conditions, allowing for the sample case. This prevents using gradient condition arguments, which makes the proofs very challenging. We also weaken the well-known sufficient condition for the uniqueness of finite-dimensional spatial quantiles.

We prove that an abelian variety whose endomorphism ring is a maximal order can be written as a direct product of simple factors with the same property, in which furthermore two isogenous factors have isomorphic nth powers for some n. Conversely every such product has a maximal order as endomorphism ring. We deduce from this some properties for arbitrary abelian varieties, in particular for almost complements of abelian subvarieties.

A consequence of de Finetti's representation theorem is that for every infinite sequence of exchangeable 0-1 random variables $(X_k)_{k\geq1}$, there exists a probability measure $\mu$ on the Borel sets of $[0,1]$ such that $\bar X_n = n^{-1} \sum_{i=1}^n X_i$ converges weakly to $\mu$. For a wide class of probability measures $\mu$ having smooth density on $(0,1)$, we give bounds of order $1/n$ with explicit constants for the Wasserstein distance between the law of $\bar X_n$ and $\mu$. This extends a recent result {by} Goldstein and Reinert \cite{goldstein2013stein} regarding the distance between the scaled number of white balls drawn in a P\'olya-Eggenberger urn and its limiting distribution. We prove also that, in the most general cases, the distance between the law of $\bar X_n$ and $\mu$ is bounded below by $1/n$ and above by $1/\sqrt{n}$ (up to some multiplicative constants). For every $\delta \in [1/2,1]$, we give an example of an exchangeable sequence such that this distance is of order $1/n^\delta$.

In this article, we construct the Hitchin fibration for groups following the scheme outlined by Frenkel-Ngo in the case of SL_{2}. This construction uses as a decisive tool the Vinberg's semigroup. The total space of Hitchin is obtained by taking the fiber product of the Hecke stack with the diagonal of the stack of G-bundles $Bun_{G}$; we prove a transversality statement between the intersection complex of the Hecke stack and the diagonal of $Bun_{G}$, over a sufficiently big open subset, in order to get local applications, such that the fundamental lemma for the spherical Hecke algebra. Along the proof of this theorem, we establish a result concerning the integral conjugacy classes of the points of a simply connected group in a local field., 44 pages, in French, improved presentation of main result after a comment by Varshavsky

Ternary sextics and quaternary quartics are the smallest cases where there exist nonnegative polynomials that are not sums of squares (SOS). A complete classification of the difference between these cones was given by G. Blekherman via analyzing the extreme rays of the corresponding dual cones. However, an exact computational approach in order to build separating extreme rays for nonnegative polynomials that are not sums of squares is a widely open problem. We provide a method substantially simplifying this computation for certain classes of polynomials on the boundary of the PSD cones. In particular, our method yields separating extreme rays for every nonnegative ternary sextic with at least seven zeros. As an application to further instances, we compute a rational certificate proving that the Motzkin polynomial is not SOS., Comment: 17 pages, 1 figure; some minor corrections and extensions

We prove existence (uniqueness is easy) of a weak solution to a boundary value problem for an equation like $(v-1)^+_t = v_{xx} + F(v)_x$ where the function $F: \Bbb R\rightarrow\Bbb R$ is only supposed to be locally lipschitz continuous. In order to replace the lack of compactness in t on v , we use nonlinear semigroup theory.

When it is planned to erect artificial windbreaks, it is necessary to estimate the stress that will be exerted by the wind in order to calculate the mechanical characteristics of the supports. To that end, this paper proposes simple fomulas to calculate the drag coefficients of artificial windbreaks, based on their geometrical characteristics. The drag force exerted by the wind on a windbreak having an height H can be written: $$D = \tfrac{1}{2}C_{dh} \rho u^2 \left( H \right) \cdot H$$ for unit length and taking the reference wind speed u(H) at H level. The drag coefficient C dh is calculated by integrating elementary drag forces between ground level and H level, which can be written as: $${\text{d}}D = \tfrac{1}{2}C_d \rho u^2 \left( z \right){\text{d}}z$$ C d being the drag coefficient of a grid element as determined in an uniform flow. Considering the logarithmic wind profile u(z)= u */k ln z/z 0 and integrating between z 0 and H leads to the following relation between C dh and C d: $$\frac{{C_{dh} }}{{C_d }} = 1 - \frac{2}{{\ln H/z_0 }} + \frac{2}{{\left( {\ln H/z_0 } \right)^2 }}$$ which is illustrated by Figure 2. The drag coefficient C d may be determined as a function of windbreak porosity φ following the works of Valensi and Rebont (1959) and Castro (1971). Valensi and Rebont propose: $$C_d = 2.\frac{{1 - m^2 \phi ^2 }}{{1 + m^2 \phi ^2 }}$$ with m = 0.95 for a metal web and 0.9 < m < 0.95 for a perforated plate (Figure 1). From Castro's work, lower C d values may be derived by adjusting his experimental values to a fourth-degree polynomial: $$C_d = 1.85 - 1.74\phi - 0.85\phi ^2 + 1.73\phi ^3 - \phi ^4$$ . Using these relationships, it is possible to calculate C dh knowing φ and H/z 0. The results of computations for H/z 0 = 200 are compared with the experimental results of Tani (1958), Hagen and Skidmore (1971), Seginer (1975) and de Bray (1971) in Figure 3. While C dh estimations based on C d values derived from Valensi and Rebont (1969) are in good agreement with experimental values of Tani, Hagen and Skidmore and Seginer, calculations derived from Castro's data agree well with de Bray's results. It is not possible at the present time to decide which is the more suitable relationship, but it seems preferable for structure computations to adopt C d values based on the work of Valensi and Rebont. The determination of the application point of the drag force leads to the relation $$\frac{{z_1 }}{H} = \frac{1}{2}\frac{{\left( {\ln H/z_0 } \right)^2 - \ln H/z_0 + \tfrac{1}{2}}}{{\left( {\ln H/z_0 } \right)^2 - 2\ln H/z_0 + 2}}$$ giving the z 1 level of this point. The variation of z 1/H as a function of H/z 0 is given in Figure 4.