By S. Kusuoka, A. Yamazaki

A lot of monetary difficulties can formulated as restricted optimizations and equilibration in their options. a number of mathematical theories were providing economists with essential machineries for those difficulties coming up in financial thought. Conversely, mathematicians were prompted through numerous mathematical problems raised by way of financial theories. The sequence is designed to compile these mathematicians who have been heavily attracted to getting new demanding stimuli from monetary theories with these economists who're looking for potent mathematical instruments for his or her researchers.

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3 a) On montre par une récurrence immédiate que : Pl(k) (X) = l+´ 2 k P (k) l+´ l−´ X+ 2 2 . l+´ |l| + 1 Or l et ´ ont le même signe. Donc | | = . 2 2 On obtient l’égalité attendue. b) Comme 1 + |l[ 1 l’égalité de 3) a) assure : |P (k) (l)| 2k |Pl(k) (1)|. D’après 2) b), on a : |P (k) (l)| c Hachette Livre – H-Prépa Exercices, Maths La photocopie non autorisée est un délit 2k Pl (k) ∞ Tn (1). l−´ x +1 x −1 l+´ X+ =l + . 2 2 2 2 x +1 Or : 0 et −1 l 1. 2 Donc : l−´ l+´ X+ x 1. −1 2 2 On en déduit la majoration de Pl ∞ par P l’inégalité : 2k P ∞ Tn(k) (1).

45 ALGÈBRE ET GÉOMÉTRIE Cette application est un endomorphisme de R[X]. Nous devons résoudre une équation linéaire. Soit k 2. E( k2 ) k f(X ) = 2 j =1 k 2j Soit a un vecteur non nul de F , b un vecteur non nul de E et F un supplémentaire de Ka dans F. Considérons l’application linéaire g déﬁnie par : X k−2 . ∀x ∈ F g(x) = 0; g(a) = b. Ce polynôme a pour degré k − 2. Cette application n’est pas nulle, mais f ◦ g ◦ f = 0. Et f(1) = f(X) = 0. Le noyau de f est Vect(1, X). L’application f est donc surjective.

IIn a) Montrer que la matrice P est inversible et donner son inverse. b) Calculer la matrice P −1 M P. c) Quelle relation en déduit-on pour Det(M) ? 2* Soit A, B, C et D quatre matrices de Mn (C). On considère la matrice de M2n (C) déﬁnie par A B M= . C D a) Montrer que si A est inversible, on a : DetM = DetA. Det(D − C A−1 B). b) On considère N = DetA DetC DetB . DetD Montrer que, si M est de rang n, la matrice N est de déterminant nul. 37 ALGÈBRE ET GÉOMÉTRIE 12 1) a) Essayer directement de déterminer l’inverse de P.