By Ambrosetti A., Felli V., Malchiodi A.

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3] MacMahon (P. ). — Combinatory Analysis, 1. — Cambridge Univ. Press,  (r´ eimpression : Chelsea Publ. , New York, ). ). — On the Foundations of Combinatorial Theory I. Theory of M¨ obius Functions, Z. Wahrscheinlichkeitstheorie, t. 2, , p. 340–368. ). — A combinatorial lemma and its appllication to probability theory, Trans. Amer. Math. , t. 82, , p. 323–339. 2 Mono¨ıde libre construit sur Z. Mono¨ıde commutatif libre construit sur Z. Homomorphisme canonique de Mo(Z) dans Ab(Z).

8 ) Raisonnant de mani`ere analogue pour les sommets 2 et 3, on trouve (5) N= b+c c+k c+a a+k a+b . /(ε1 ! ε2 ! ε3 ! ε4 ! ) ; si l’on ´evalue les εi conform´ement aux formules (1) et (2) et qu’on somme sur l’entier n soumis a ` la condition (3), on trouve p (6) N= n=|k| (a + b + c − n)! (a − n)! (b − n)! (c − n)! (n + k)! (n − k)! et la comparaison de (5) et (6) ´etablit la formule (A). ´ 3. Etude des circuits impairs L’entier ε4 + ε5 ´etant cette fois impair, il faut remplacer les formules (2) et (3) par les suivantes : (2 ) (3 ) κ≤n≤p ε4 = n + k, ε5 = n − k + 1, avec κ = max{−k, k − 1} et p = min{a, b, c}.

3 ! ε4 ! ) ; si l’on ´evalue les εi conform´ement aux formules (1) et (2) et qu’on somme sur l’entier n soumis a ` la condition (3), on trouve p (6) N= n=|k| (a + b + c − n)! (a − n)! (b − n)! (c − n)! (n + k)! (n − k)! et la comparaison de (5) et (6) ´etablit la formule (A). ´ 3. Etude des circuits impairs L’entier ε4 + ε5 ´etant cette fois impair, il faut remplacer les formules (2) et (3) par les suivantes : (2 ) (3 ) κ≤n≤p ε4 = n + k, ε5 = n − k + 1, avec κ = max{−k, k − 1} et p = min{a, b, c}.

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Ground states of nonlinear Schrodinger equations with potentials vanishing at infinity by Ambrosetti A., Felli V., Malchiodi A.


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