In this paper we provide a complete characterization of fully nonlinear conformally invariant differential operators of any integer order on R, which extends the result proved for operators of the second order by A. Li and the first named author in [38]. In particular we prove existence and uniqueness of a family of tensors (suitably invariant under Möbius transformations) which are the basic b...

Supervisory control and fault diagnosis of hybrid systems need to have complete information about the discrete states transitions of the underling system. From this point of view, the hybrid system should be abstracted to a Discrete Trace Transition System (DTTS) and represented by a discrete mode transition graph. In this paper an effective method is proposed for generating discrete mode trans...

We present a formalism for dimensional reduction based on the local properties of invariant cross-sections (“fields”) and differential operators. This formalism does not need an ansatz for the invariant fields and is convenient when the reducing group is non-compact. In the approach presented here, splittings of some exact sequences of vector bundles play a key role. In the case of invariant fi...

Point transformations of the 3-rd order ordinary differential equations are considered. Special classes of ordinary differential equations that are invariant under the general point transformations are constructed.

A new version of differential renormalization is presented. It is based on pulling out certain differential operators and introducing a logarithmic dependence into diagrams. It can be defined either in coordinate or momentum space, the latter being more flexible for treating tadpoles and diagrams where insertion of counterterms generates tadpoles. Within this version, gauge invariance is automa...

We present a generalization of Lie’s method for finding the group invariant solutions to a system of partial differential equations. Our generalization relaxes the standard transversality assumption and encompasses the common situation where the reduced differential equations for the group invariant solutions involve both fewer dependent and independent variables. The theoretical basis for our ...

A frequently studied problem of geometric singular perturbation theory consists in establishing the presence of trajectories of certain types (homoclinic, heteroclinic, satisfying given boundary conditions, etc.) approximating singular ones for the unperturbed problem. A useful tool for this problem has been established in Jones et al. [2] and called “Exchange Lemma” by the authors. It resemble...

Let Fλ be the space of tensor densities on R n of degree λ (or, equivalently, of conformal densities of degree −λn) considered as a module over the Lie algebra o(p+1, q+1). We classify o(p+1, q+1)-invariant bilinear differential operators from Fλ ⊗ Fμ to Fν . The classification of linear o(p + 1, q + 1)invariant differential operators from Fλ to Fμ already known in the literature (see [6, 9]) i...

The goal of this paper is to describe, in as much detail as possible, the structure of the algebra of differential invariants of a Lie pseudo-group. Under the assumption of local freeness of the prolonged pseudo-group action, we develop algorithms for locating a finite generating set of differential invariants, establishing the recurrence relations for the differentiated invariants, and fixing ...