Non-integrability by discrete quadratures

2014 ◽  
Vol 2014 (687) ◽  
Author(s):  
Guy Casale ◽  
Julien Roques
Keyword(s):  
Difference Equations ◽  
Galois Groups ◽  
Necessary Condition ◽  
Variational Equations ◽  
Algebraic Solutions

Abstract.We give a necessary condition for integrability by discrete quadratures of systems of difference equations: the discrete variational equations along algebraic solutions must have virtually solvable Galois groups. This necessary condition à la Morales and Ramis is used in order to prove that

scholarly journals Extensions of differential representations of SL2 and tori

2012 ◽  
Vol 12 (1) ◽  
pp. 199-224 ◽  
Author(s):  
Andrey Minchenko ◽  
Alexey Ovchinnikov
Keyword(s):  
Irreducible Representation ◽  
Representation Theory ◽  
Difference Equations ◽  
Algebraic Groups ◽  
Galois Groups ◽  
Irreducible Representations ◽  
Direct Sums ◽  
Rational Representation ◽  
Linear Differential ◽  
Linear Differential Algebraic Groups

AbstractLinear differential algebraic groups (LDAGs) measure differential algebraic dependencies among solutions of linear differential and difference equations with parameters, for which LDAGs are Galois groups. Differential representation theory is a key to developing algorithms computing these groups. In the rational representation theory of algebraic groups, one starts with ${\mathbf{SL} }_{2} $ and tori to develop the rest of the theory. In this paper, we give an explicit description of differential representations of tori and differential extensions of irreducible representation of ${\mathbf{SL} }_{2} $. In these extensions, the two irreducible representations can be non-isomorphic. This is in contrast to differential representations of tori, which turn out to be direct sums of isotypic representations.

scholarly journals A stability condition for nth order difference equations

1971 ◽  
Vol 12 (1) ◽  
pp. 24-30
Author(s):  
Russell A. Smith
Keyword(s):  
Difference Equations ◽  
Hermitian Form ◽  
Real Variable ◽  
Necessary Condition ◽  
Quadratic Lyapunov Function ◽  
Globally Asymptotically Stable ◽  
Order Difference ◽  
Constant Coefficients ◽  
Image Position ◽  
Special Case

Consider the system of difference equationsin which the unknown x(t) is a complex m-vector, t is a real variable and a1, …, an are complex m × m matrices whose elements are functions of t, x(t), x(t+1), …, x(t+n – 1). A positive definite hermitian form V(x1x2, …, xn), with constant coefficients, is called a strong autonomous quadratic Lyapunov function (written strong AQLF) of (1) if there exists a constant K > 1 such that K2v(t+1) < v(t) for all non-zero solutions x(t)of (1), where v(t) = V(x(t), x(t+ 1), …, x(t+n —1)). The existence of a strong AQLF is a sufficient condition for the trivial solution x =0 of (1) to be globally asymptotically stable. It is a necessary condition only in the special case of an equation

scholarly journals On classical irregular q-difference equations

2012 ◽  
Vol 148 (5) ◽  
pp. 1624-1644 ◽  
Author(s):  
Julien Roques
Keyword(s):  
Differential Equations ◽  
Difference Equations ◽  
Galois Groups ◽  
Physical Literature ◽  
Hypergeometric Equations

AbstractThe primary aim of this paper is to (provide tools to) compute Galois groups of classical irregular q-difference equations. We are particularly interested in quantizations of certain differential equations that arise frequently in the mathematical and physical literature, namely confluent generalized q-hypergeometric equations and q-Kloosterman equations.

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