The complex quadratic form , where z is a fixed vector in and is its transpose, and P is any permutation matrix, is shown to be a convex combination of the quadratic forms , where 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 .
Nous montrons que la forme quadratique complexe , où z est un vecteur donné dans et est son transposé, et P est une matrice de permutation, est une combinaison convexe des formes quadratiques , où les sont des matrices de permutation symétriques. On en déduit que la densité de probabilité optimale associée à l'indice chiral d'un échantillon d'une distribution bivariée est symétrique. Ce résultat est utilisé pour localiser la borne supérieure de l'indice chiral d'une distribution bivariée quelconque dans l'intervalle .
Accepted:
Published online:
Don Coppersmith 1; Michel Petitjean 2
@article{CRMATH_2005__340_8_599_0, author = {Don Coppersmith and Michel Petitjean}, title = {About the optimal density associated to the chiral index of a sample from a bivariate distribution}, journal = {Comptes Rendus. Math\'ematique}, pages = {599--604}, publisher = {Elsevier}, volume = {340}, number = {8}, year = {2005}, doi = {10.1016/j.crma.2005.03.011}, language = {en}, }
TY - JOUR AU - Don Coppersmith AU - Michel Petitjean TI - About the optimal density associated to the chiral index of a sample from a bivariate distribution JO - Comptes Rendus. Mathématique PY - 2005 SP - 599 EP - 604 VL - 340 IS - 8 PB - Elsevier DO - 10.1016/j.crma.2005.03.011 LA - en ID - CRMATH_2005__340_8_599_0 ER -
Don Coppersmith; Michel Petitjean. About the optimal density associated to the chiral index of a sample from a bivariate distribution. Comptes Rendus. Mathématique, Volume 340 (2005) no. 8, pp. 599-604. doi : 10.1016/j.crma.2005.03.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.03.011/
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