About the optimal density associated to the chiral index of a sample from a bivariate distribution

The complex quadratic form $z^{\prime }Pz$, where z is a fixed vector in ${\mathbf{C}}^n$ and $z^{\prime }$ is its transpose, and P is any permutation matrix, is shown to be a convex combination of the quadratic forms $z^{\prime }{P}_{\sigma }z$, where $P_{\sigma }$ denotes the symmetric permutation matrices. We deduce that the optimal probability density associated to the chiral index of a sample from a bivariate distribution is symmetric. This result is used to locate the upper bound of the chiral index of any bivariate distribution in the interval $[1?1/\pi ,1?1/2\pi ]$. To cite this article: D. Coppersmith, M. Petitjean, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

The division problem for tempered distributions of one variable

We give a characterization, in one variable case, of those $C^{\infty }$ multipliers F such that the division problem is solvable in $S^{\prime }(\mathbb{R})$. For these functions $F\in O_M(\mathbb{R})$ we even prove that the multiplication operator $M_F(G)=FG$ has a continuous linear right inverse on $S^{\prime }(\mathbb{R})$, in contrast to what happens in the several variables case, as was shown by Langenbruch.     

On the geography of threefolds of general type

Let X be a complex nonsingular projective 3-fold of general type. We show that there are positive constants c, $c^{\prime }$ and $m_1$ such that $\chi ({\omega }_X)??c\mathrm{Vol}(X)$ and $P_m(X)?c^{\prime }{m}^3\mathrm{Vol}(X)$ for all $m?m_1$.     

N-cyclic functions and multiple subharmonic solutions of Duffingʼs equation

We introduce, in the abstract framework of finite isometry groups on a Hilbert space, a generalization of antiperiodicity called N-cyclicity. The non-existence of N-cyclic solutions of a certain type for the autonomous ODE $x^{″}+g(x)=0$ implies the existence of N different subharmonic solutions for some forced equations of the type $x^{″}+g(x)+c{x}^{\prime }=\epsilon f(t)$ where c and ε are some positive constants and f is, for instance, a sinusoidal function.