### A Characterization of Linear Difference Equations Which are Solvable by Elementary Operations.

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We obtain an algebraic interpretation by means of the Picard-Vessiot theory of a result by Ziglin about the self-intersection of complex separatrices of time-periodically perturbed one-degree of freedom complex analytical Hamiltonian systems.

Let A be a commutative algebra without zero divisors over a field k. If A is finitely generated over k, then there exist well known characterizations of all k-subalgebras of A which are rings of constants with respect to k-derivations of A. We show that these characterizations are not valid in the case when the algebra A is not finitely generated over k.

If $X$ is a smooth scheme over a perfect field of characteristic $p$, and if ${\mathcal{D}}_{X}^{\left(\infty \right)}$ is the sheaf of differential operators on $X$ [7], it is well known that giving an action of ${\mathcal{D}}_{X}^{\left(\infty \right)}$ on an ${\mathcal{O}}_{X}$-module $\mathcal{E}$ is equivalent to giving an infinite sequence of ${\mathcal{O}}_{X}$-modules descending $\mathcal{E}$ via the iterates of the Frobenius endomorphism of $X$ [5]. We show that this result can be generalized to any infinitesimal deformation $f:X\to S$ of a smooth morphism in characteristic $p$, endowed with Frobenius liftings. We also show that it extends to adic...

A closed loop parametrical identification procedure for continuous-time constant linear systems is introduced. This approach which exhibits good robustness properties with respect to a large variety of additive perturbations is based on the following mathematical tools: (1) module theory; (2) differential algebra; (3) operational calculus. Several concrete case-studies with computer simulations demonstrate the efficiency of our on-line identification scheme.