We discuss the structure of shock singularities of the Burgers-Hopf hierarchy. It is shown that the set of singular solutions defines a stratification of the affine space of the flow parameters in the hierarchy. The stratification is associated with the Birkhoff decomposition of the Grassmannian given by the set of linear spaces spanned by the hierarchy. We then construct integrable hierarchy o...

We consider the factorization of an all-pass matrix function E(s) using proper stable minimum-phase factors. State-space formulae for the stable minimum-phase factors of E(s) are derived (when they exist), as well as state-space formulae for the Wiener-Hopf factors. This is achieved using the state-space characterization of allpass matrix functions given by Glover (1984) in conjunction with the...

The fundamental group and rational cohomology of the configuration spaces of the Skyrme and Faddeev-Hopf models are computed. Physical space is taken to be a compact oriented 3-manifold, either with or without a marked point representing an end at infinity. For the Skyrme model, the codomain is any Lie group, while for the Faddeev-Hopf model it is S2. It is determined when the topology of confi...

We construct static and time dependent exact soliton solutions for a theory of scalar fields taking values on a wide class of two dimensional target spaces, and defined on the four dimensional space-time S 3 × IR. The construction is based on an ansatz built out of special coordinates on S 3. The requirement for finite energy introduces boundary conditions that determine an infinite discrete sp...

We introduce the categories of infinitesimal Hopf modules and bimodules over an infinitesimal bialgebra. We show that they correspond to modules and bimodules over the infinitesimal version of the double. We show that there is a natural, but non-obvious way to construct a pre-Lie algebra from an arbitrary infinitesimal bialgebra and a dendriform algebra from a quasitriangular infinitesimal bial...

We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical (α < 1/2) dissipation (−∆). This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical (α = 1/2) QG equation [2]. Their approach successively increases the regularity levels of Leray-Hopf weak solutions: from L to L∞, f...

We introduce ∗-structures on braided groups and braided matrices. Using this, we show that the quantum double D(Uq(su2)) can be viewed as the quantum algebra of observables of a quantum particle moving on a hyperboloid in q-Minkowski space (a three-sphere in the Lorentz metric), and with the role of angular momentum played by Uq(su2). This provides a new example of a quantum system whose algebr...