We consider equations of the form Bf=g, where B is a Galois connection between lattices of functions. This includes the case where B is the Fenchel transform, or more generally a Moreau conjugacy. We characterize the existence and uniqueness of a solution f in terms of generalized subdifferentials, which extends K. Zimmermann's covering theorem for max-plus linear equations.
On considère des équations de la forme Bf=g, où B est une correspondance de Galois entre des treillis de fonctions, ce qui inclut le cas où B est la transformation de Fenchel, ou plus généralement une conjugaison de Moreau. Nous caractérisons l'existence et l'unicité d'une solution f, en termes de sous-différentiels généralisés, et étendons ainsi le théorème de couverture de K. Zimmermann pour les équations linéaires max-plus.
Accepted:
Published online:
Marianne Akian 1; Stéphane Gaubert 1; Vassili Kolokoltsov 2, 3
@article{CRMATH_2002__335_11_883_0, author = {Marianne Akian and St\'ephane Gaubert and Vassili Kolokoltsov}, title = {Invertibility of functional {Galois} connections}, journal = {Comptes Rendus. Math\'ematique}, pages = {883--888}, publisher = {Elsevier}, volume = {335}, number = {11}, year = {2002}, doi = {10.1016/S1631-073X(02)02594-3}, language = {en}, }
TY - JOUR AU - Marianne Akian AU - Stéphane Gaubert AU - Vassili Kolokoltsov TI - Invertibility of functional Galois connections JO - Comptes Rendus. Mathématique PY - 2002 SP - 883 EP - 888 VL - 335 IS - 11 PB - Elsevier DO - 10.1016/S1631-073X(02)02594-3 LA - en ID - CRMATH_2002__335_11_883_0 ER -
Marianne Akian; Stéphane Gaubert; Vassili Kolokoltsov. Invertibility of functional Galois connections. Comptes Rendus. Mathématique, Volume 335 (2002) no. 11, pp. 883-888. doi : 10.1016/S1631-073X(02)02594-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02594-3/
[1] Densities of idempotent measures and large deviations, Trans. Amer. Math. Soc., Volume 351 (1999) no. 11, pp. 4515-4543
[2] M. Akian, S. Gaubert, V. Kolokoltsov, Invertibility of functional Galois connections and large deviations, 2002, in preparation
[3] Synchronization and Linearity: An Algebra for Discrete Events Systems, Wiley, New York, 1992
[4] Lattice Theory, Colloq. Publ., 25, American Mathematical Society, Providence, 1995
[5] Strong regularity of matrices – a survey of results, Discrete Appl. Math., Volume 48 (1994), pp. 45-68
[6] Simple image set of (max, +) linear mappings, Discrete Appl. Math., Volume 105 (2000) no. 1–3, pp. 73-86
[7] Minimax Algebra, Lecture Notes in Econom. Math. Systems, 166, Springer, 1979
[8] Large Deviations Techniques and Applications, Jones and Barlett, Boston, MA, 1993
[9] Graphes, dioı̈des et semi-anneaux, TEC & DOC, Paris, 2001
[10] V. Kolokoltsov, On linear, additive, and homogeneous operators, 1992, in [12]
[11] Idempotent Analysis and Applications, Kluwer Academic, 1997
[12] Idempotent Analysis (V. Maslov; S. Samborskiı̆, eds.), Adv. Soviet Math., 13, American Mathematical Society, RI, 1992
[13] Quasi-continuity, Real Anal. Exchange, Volume 14 (1988/89) no. 2, pp. 259-306
[14] Mass Transportation Problems, Vol. I: Theory, Springer, 1998
[15] Convex Analysis, Princeton University Press, Princeton, NJ, 1970
[16] Variational Analysis, Springer-Verlag, Berlin, 1998
[17] Abstract Convex Analysis, Wiley, New York, 1997
[18] Further applications of the additive min-type coupling function, Optimization, Volume 51 (2002), pp. 471-485
[19] K. Zimmermann, Extremálnı́ Algebra, Ekonomický ùstav C̆SAV, Praha, 1976 (in Czech)
Cited by Sources:
Comments - Policy