Reduction is a process that uses symmetry to lower the order of a Hamiltonian system. The new variables in the reduced picture are often not canonical: there are no clear variables representing positions and momenta, and the Poisson bracket obtained is not of the canonical type. Specifically, we give two examples that give rise to brackets of the noncanonical Lie-Poisson form: the rigid body an...

Symmetry (similarity under transformation) in sensorimotor data streams can be used to derive and represent concepts from percepts, and exploit them in a cognitive framework. This entails finding mechanisms to extract symmetries out of sensorimotor data and convert them into intuitive representations; ”intuitive” implying that the representations created should have a semantic meaning to them. ...

Proper extensions that are “injective on L-related idempotents” of R-unipotent semigroups, and much more generally of the class of generalised left restriction semigroups possessing the ample and congruence conditions, referred to here as glrac semigroups, are described as certain subalgebras of a λ-semidirect product of a left regular band by an R-unipotent or by a glrac semigroup, respectivel...

Continuous wavelet transforms arising from the quasiregular representation of a semidirect product group G = R ⋊ H have been studied by various authors. Recently the attention has shifted from the irreducible case to include more general dilation groups H , for instance cyclic (more generally: discrete) or one-parameter groups. These groups do not give rise to irreducible square-integrable repr...

We give a general description of the free proonite semigroups over a semidirect product of pseudovarieties. More precisely, A (V W) is described as a closed subsemigroup of a proonite semidirect product of the form A WA V A W. As a particular case, the free proonite semigroup over J 1 V is described in terms of the geometry of the Cayley graph of the free proonite semigroup over V (here J 1 is ...

A classification scheme for regular languages or finite semigroups was proposed by Pin through tree hierarchies, a scheme related to the concatenation product, an operation on languages, and to the Schützenberger product, an operation on semigroups. Starting with a variety of finite semigroups (or pseudovariety of semigroups) V, a pseudovariety of semigroups ◊u(V) is associated to each tree u. ...

We consider the distance enumerator ∆G(x) of a finite permutation group G, which is the polynomial ∑ g∈G xn−π(g), where n is the degree of G and π(g) the number of fixed points of g ∈ G. In particular, we introduce a bivariate polynomial which is a special case of the cycle index of G, and from which ∆G(x) can be obtained, and then use this new polynomial to prove some identities relating the d...

We develop a multi-scale theory for group of diffeomorphisms based on previous works [BGBHR11, RVW11, SNLP11, SLNP11]. The purpose of the paper is to develop in details a variational approach for multiscale on diffeomorphisms and to generalize to several scales the semidirect product of group representation. We also show that the approaches presented in [SNLP11, SLNP11] and the mixture of kerne...