Linearizing the Word Problem in (some) Free Fields

We describe a solution of the word problem in free fields (coming from non-commutative polynomials over a commutative field) using elementary linear algebra, provided that the elements are given by minimal linear representations. It relies on the normal form of Cohn and Reutenauer and can be used more generally to (positively) test rational identities. Moreover we provide a construction of minimal linear representations for the inverse of non-zero elements.Comment: 22 pages, slightly updated, accepted in IJA

Breakdown of the standard Perturbation Theory and Moving Boundary Approximation for "Pulled" Fronts

The derivation of a Moving Boundary Approximation or of the response of a coherent structure like a front, vortex or pulse to external forces and noise, is generally valid under two conditions: the existence of a separation of time scales of the dynamics on the inner and outer scale and the existence and convergence of solvability type integrals. We point out that these conditions are not satisfied for pulled fronts propagating into an unstable state: their relaxation on the inner scale is power law like and in conjunction with this, solvability integrals diverge. The physical origin of this is traced to the fact that the important dynamics of pulled fronts occurs in the leading edge of the front rather than in the nonlinear internal front region itself. As recent work on the relaxation and stochastic behavior of pulled fronts suggests, when such fronts are coupled to other fields or to noise, the dynamical behavior is often qualitatively different from the standard case in which fronts between two (meta)stable states or pushed fronts propagating into an unstable state are considered.Comment: pages Latex, submitted to a special issue of Phys. Rep. in dec. 9

Flat systems, equivalence and trajectory generation

Flat systems, an important subclass of nonlinear control systems introduced via differential-algebraic methods, are defined in a differential geometric framework. We utilize the infinite dimensional geometry developed by Vinogradov and coworkers: a control system is a diffiety, or more precisely, an ordinary diffiety, i.e. a smooth infinite-dimensional manifold equipped with a privileged vector field. After recalling the definition of a Lie-Backlund mapping, we say that two systems are equivalent if they are related by a Lie-Backlund isomorphism. Flat systems are those systems which are equivalent to a controllable linear one. The interest of such an abstract setting relies mainly on the fact that the above system equivalence is interpreted in terms of endogenous dynamic feedback. The presentation is as elementary as possible and illustrated by the VTOL aircraft

Some applications of p-adic uniformization to algebraic dynamics

This is not a research paper, but a survey submitted to a proceedings volume.Comment: 21 pages, LaTe

Dynamics of a Disoriented Chiral Condensate

We use the linear $\sigma$ model to analyse the dynamics of a disoriented chiral condensate. For idealized boundary conditions appropriate to high energy collisions, the problem can be reduced to a one dimensional one. The evolution of the chiral state is then that of a simple dynamical system and can be studied analytically.Comment: 14 pages Latex, LPTHE Orsay 94/18 , SPhT T94/01

Spin connection as Lorentz gauge field: propagating torsion

We propose a modified gravitational action containing besides the Einstein-Cartan term some quadratic contributions resembling the Yang-Mills lagrangian for the Lorentz spin connections. We outline how a propagating torsion arises and we solve explicitly the linearised equations of motion on a Minkowski background. We identify among torsion components six degrees of freedom: one is carried by a pseudo-scalar particle, five by a tachyon field. By adding spinor fields and neglecting backreaction on the geometry, we point out how only the pseudo-scalar particle couples directly with fermions, but the resulting coupling constant is suppressed by the ratio between fermion and Planck masses. Including backreaction, we demonstrate how the tachyon field provides causality violation in the matter sector, via an interaction mediated by gravitational waves.Comment: 7 pages, no figures, new section adde