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ON CONSIDERE UNE EQUATION DIFFERENTIELLE, L, D'ORDRE N A COEFFICIENTS DANS LE CORPS K DES FONCTIONS RATIONNELLES A COEFFICIENTS DANS C, ET UNE MATRICE FONDAMENTALE DE SOLUTIONS U. ON ETUDIE ALORS LES RELATIONS ALGEBRIQUES, A COEFFICIENTS DANS K, ENTRE LES ELEMENTS D'UNE PARTIE Y DE U. IL Y A DE NOMBREUSES MOTIVATIONS A L'ETUDE DE CE PROBLEME, NOTAMMENT ARITHMETIQUE, VIA LE THEOREME DE SIEGEL-SHIDLOVSKII. LE PREMIER CHAPITRE ETUDIE LE CAS OU Y EST LA PREMIERE LIGNE DE U. DANS LE PROLONGEMENT DES TRAVAUX DE FANO, PUIS SINGER, ON OBTIENT UN CRITERE DE RESOLUBILITE DE L'EQUATION DIFFERENTIELLE L, EN TERMES D'EQUATIONS D'ORDRE INFERIEUR A N. DANS LE SECOND CHAPITRE Y EST UNE COLONNE DE U ET L UN OPERATEUR HYPERGEOMETRIQUE. ON CALCULE ALORS LE DEGRE DE TRANSCENDANCE, SUR K, DU CORPS DIFFERENTIEL, ENGENDRE SUR K, PAR TOUTE SOLUTION DE L (AMELIORANT AINSI DES MAJORATIONS DE SALICHOV). ON EN DEDUIT DES RESULTATS DE THEORIE DES NOMBRES, VIA LE THEOREME DE SIEGEL-SHIDLOVSKII. DANS LE TROISIEME ET DERNIER CHAPITRE ON SUPPOSE CONNU LE GROUPE DE GALOIS DIFFERENTIEL, G, DE L (C'EST LE CAS LORSQUE L EST HYPERGEOMETRIQUE). EN SUPPOSANT G REDUCTIF ET UNIMODULAIRE, ON DONNE UN ALGORITHME DE CALCUL DE L'IDEAL DES RELATIONS ALGEBRIQUES A COEFFICIENTS DANS K LIANT TOUS LES ELEMENTS DE LA MATRICE U (ICI U EST EGAL A Y)
Book Description
ON CONSIDERE UNE EQUATION DIFFERENTIELLE, L, D'ORDRE N A COEFFICIENTS DANS LE CORPS K DES FONCTIONS RATIONNELLES A COEFFICIENTS DANS C, ET UNE MATRICE FONDAMENTALE DE SOLUTIONS U. ON ETUDIE ALORS LES RELATIONS ALGEBRIQUES, A COEFFICIENTS DANS K, ENTRE LES ELEMENTS D'UNE PARTIE Y DE U. IL Y A DE NOMBREUSES MOTIVATIONS A L'ETUDE DE CE PROBLEME, NOTAMMENT ARITHMETIQUE, VIA LE THEOREME DE SIEGEL-SHIDLOVSKII. LE PREMIER CHAPITRE ETUDIE LE CAS OU Y EST LA PREMIERE LIGNE DE U. DANS LE PROLONGEMENT DES TRAVAUX DE FANO, PUIS SINGER, ON OBTIENT UN CRITERE DE RESOLUBILITE DE L'EQUATION DIFFERENTIELLE L, EN TERMES D'EQUATIONS D'ORDRE INFERIEUR A N. DANS LE SECOND CHAPITRE Y EST UNE COLONNE DE U ET L UN OPERATEUR HYPERGEOMETRIQUE. ON CALCULE ALORS LE DEGRE DE TRANSCENDANCE, SUR K, DU CORPS DIFFERENTIEL, ENGENDRE SUR K, PAR TOUTE SOLUTION DE L (AMELIORANT AINSI DES MAJORATIONS DE SALICHOV). ON EN DEDUIT DES RESULTATS DE THEORIE DES NOMBRES, VIA LE THEOREME DE SIEGEL-SHIDLOVSKII. DANS LE TROISIEME ET DERNIER CHAPITRE ON SUPPOSE CONNU LE GROUPE DE GALOIS DIFFERENTIEL, G, DE L (C'EST LE CAS LORSQUE L EST HYPERGEOMETRIQUE). EN SUPPOSANT G REDUCTIF ET UNIMODULAIRE, ON DONNE UN ALGORITHME DE CALCUL DE L'IDEAL DES RELATIONS ALGEBRIQUES A COEFFICIENTS DANS K LIANT TOUS LES ELEMENTS DE LA MATRICE U (ICI U EST EGAL A Y)
Author: Vladimir Dorodnitsyn Publisher: CRC Press ISBN: 9781420083101 Category : Mathematics Languages : en Pages : 344
Book Description
Intended for researchers, numerical analysts, and graduate students in various fields of applied mathematics, physics, mechanics, and engineering sciences, Applications of Lie Groups to Difference Equations is the first book to provide a systematic construction of invariant difference schemes for nonlinear differential equations. A guide to methods
Author: Ellis Robert Kolchin Publisher: American Mathematical Soc. ISBN: 9780821805428 Category : Mathematics Languages : en Pages : 660
Book Description
The work of Joseph Fels Ritt and Ellis Kolchin in differential algebra paved the way for exciting new applications in constructive symbolic computation, differential Galois theory, the model theory of fields, and Diophantine geometry. This volume assembles Kolchin's mathematical papers, contributing solidly to the archive on construction of modern differential algebra. This collection of Kolchin's clear and comprehensive papers--in themselves constituting a history of the subject--is an invaluable aid to the student of differential algebra. In 1910, Ritt created a theory of algebraic differential equations modeled not on the existing transcendental methods of Lie, but rather on the new algebra being developed by E. Noether and B. van der Waerden. Building on Ritt's foundation, and deeply influenced by Weil and Chevalley, Kolchin opened up Ritt theory to modern algebraic geometry. In so doing, he led differential geometry in a new direction. By creating differential algebraic geometry and the theory of differential algebraic groups, Kolchin provided the foundation for a "new geometry" that has led to both a striking and an original approach to arithmetic algebraic geometry. Intriguing possibilities were introduced for a new language for nonlinear differential equations theory. The volume includes commentary by A. Borel, M. Singer, and B. Poizat. Also Buium and Cassidy trace the development of Kolchin's ideas, from his important early work on the differential Galois theory to his later groundbreaking results on the theory of differential algebraic geometry and differential algebraic groups. Commentaries are self-contained with numerous examples of various aspects of differential algebra and its applications. Central topics of Kolchin's work are discussed, presenting the history of differential algebra and exploring how his work grew from and transformed the work of Ritt. New directions of differential algebra are illustrated, outlining important current advances. Prerequisite to understanding the text is a background at the beginning graduate level in algebra, specifically commutative algebra, the theory of field extensions, and Galois theory.
Author: Ranis Ibragimov Publisher: Walter de Gruyter GmbH & Co KG ISBN: 3111387801 Category : Mathematics Languages : en Pages : 372
Book Description
The book is focused on physical interpretation and visualization of the obtained invariant solutions for nonlinear mathematical modeling of atmospheric and ocean waves. This volume represents a unique blend of analytical and numerical methods complemented by the author's developments in ocean and atmospheric sciences and it is meant for researchers and graduate students interested in applied mathematics and mathematical modeling.
Author: Roger Chalkley Publisher: American Mathematical Soc. ISBN: 0821839918 Category : Mathematics Languages : en Pages : 386
Book Description
The problem of deducing the basic relative invariants possessed by monic homogeneous linear differential equations of order $m$ was initiated in 1879 with Edmund Laguerre's success for the special case $m = 3$. It was solved in number 744 of the Memoirs of the AMS (March 2002), by a procedure that explicitly constructs, for any $m \geq3$, each of the $m - 2$ basic relative invariants. During that 123-year time span, only a few results were published about the basic relative invariants for other classes of ordinary differential equations. With respect to any fixed integer $\, m \geq 1$, the author begins by explicitly specifying the basic relative invariants for the class $\, \mathcal{C {m,2 $ that contains equations like $Q {m = 0$ in which $Q {m $ is a quadratic form in $y(z), \, \dots, \, y{(m) (z)$ having meromorphic coefficients written symmetrically and the coefficient of $\bigl( y{(m) (z) \bigr){2 $ is $1$.Then, in terms of any fixed positive integers $m$ and $n$, the author explicitly specifies the basic relative invariants for the class $\, \mathcal{C {m, n $ that contains equations like $H {m, n = 0$ in which $H {m, n $ is an $n$th-degree form in $y(z), \, \dots, \, y{(m) (z)$ having meromorphic coefficients written symmetrically and the coefficient of $\bigl( y{(m) (z) \bigr){n $ is $1$.These results enable the author to obtain the basic relative invariants for additional classes of ordinary differential equa
Author: Elie Compoint
Author: Elie Compoint
Author: Elie Compoint
Author: Ernest Vessiot
Author: Elie Cartan
Author: Michel Philippoff
Author: Vladimir Dorodnitsyn
Author: Ellis Robert Kolchin
Author: Ranis Ibragimov
Author: Roger Chalkley
Author: Emile Picard