Generalized Dynkin diagrams and root systems and their folding

Graphs which generalize the simple or affine Dynkin diagrams are introduced. Each diagram defines a bilinear form on a root system and thus a reflection group. We present some properties of these groups and of their natural "Coxeter element". The folding of these graphs and groups is also discussed, using the theory of C-algebras. (Proceedings of the Taniguchi Symposium {Topological Field Theory, Primitive Forms and Related Topics}, Kyoto Dec 1996)Comment: plain tex, 7 figure

Inverse polynomial optimization

We consider the inverse optimization problem associated with the polynomial program f^*=\min \{f(x): x\in K\}$and a given current feasible solution$y\in K$. We provide a systematic numerical scheme to compute an inverse optimal solution. That is, we compute a polynomial$\tilde{f}$(which may be of same degree as$f$if desired) with the following properties: (a)$y$is a global minimizer of$\tilde{f}$on$K$with a Putinar's certificate with an a priori degree bound$d$fixed, and (b),$\tilde{f}$minimizes$\Vert f-\tilde{f}\Vert$(which can be the$\ell_1$,$\ell_2$or$\ell_\infty$-norm of the coefficients) over all polynomials with such properties. Computing$\tilde{f}_d$reduces to solving a semidefinite program whose optimal value also provides a bound on how far is$f(\y)$from the unknown optimal value$f^*$. The size of the semidefinite program can be adapted to the computational capabilities available. Moreover, if one uses the$\ell_1$-norm, then$\tilde{f}$ takes a simple and explicit canonical form. Some variations are also discussed.Comment: 25 pages; to appear in Math. Oper. Res; Rapport LAAS no. 1114

Level sets and non Gaussian integrals of positively homogeneous functions

We investigate various properties of the sublevel set $\{x \,:\,g(x)\leq 1\}$ and the integration of $h$ on this sublevel set when $g$ and $h$are positively homogeneous functions. For instance, the latter integral reduces to integrating $h\exp(-g)$ on the whole space $R^n$ (a non Gaussian integral) and when $g$ is a polynomial, then the volume of the sublevel set is a convex function of the coefficients of $g$. In fact, whenever $h$ is nonnegative, the functional $\int \phi(g(x))h(x)dx$ is a convex function of $g$ for a large class of functions $\phi:R_+\to R$. We also provide a numerical approximation scheme to compute the volume or integrate $h$ (or, equivalently to approximate the associated non Gaussian integral). We also show that finding the sublevel set $\{x \,:\,g(x)\leq 1\}$ of minimum volume that contains some given subset $K$ is a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussian-like property of non Gaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function

On the Counting of Fully Packed Loop Configurations. Some new conjectures

New conjectures are proposed on the numbers of FPL configurations pertaining to certain types of link patterns. Making use of the Razumov and Stroganov Ansatz, these conjectures are based on the analysis of the ground state of the Temperley-Lieb chain, for periodic boundary conditions and so-called ``identified connectivities'', up to size $2n=22$