Virtual braids from a topological viewpoint

Virtual braids are a combinatorial generalization of braids. We present abstract braids as equivalence classes of braid diagrams on a surface, joining two distinguished boundary components. They are identified up to isotopy, compatibility, stability and Reidemeister moves. We show that virtual braids are in a bijective correspondence with abstract braids. Finally we demonstrate that for any abstract braid, its representative of minimal genus is unique up to compatibility and Reidemeister moves. The genus of such a representative is thus an invariant for virtual braids. We also give a complete proof of the fact that there is a bijective correspondence between virtually equivalent virtual braid diagrams and braid-Gauss diagrams.

Operators having the symmetrized bidisc as a spectral set

We characterize those commuting pairs of operators on Hubert space that have the symmetrized bidisc as a spectral set in terms of the positivity of a hermitian operator pencil (without any assumption about the joint spectrum of the pair). Further equivalent conditions are that the pair has a normal dilation to the distinguished boundary of the symmetrized bidisc, and that the pair has the symmetrized bidisc as a complete spectral set. A consequence is that every contractive representation of the operator algebra A(Γ) of continuous functions on the symmetrized bidisc analytic in the interior is completely contractive. The proofs depend on a polynomial identity that is derived with the aid of a realization formula for doubly symmetric hereditary polynomials, which are positive on commuting pairs of contractions.

Slice Maps and Multipliers of Invariant Subspaces

Let be the closed bidisc and T2 be its distinguished boundary. For be a slice map, that is, and Then ker Φαβ is an invariant subspace, and it is not difficult to describe ker Φαβ and In this paper, we study the set of all multipliers for an invariant subspace M such that the common zero set of M contains that of ker Φαβ.

A Plessner Decomposition Along Transverse Curves

A classical theorem of Plessner [6] asserts that any holomorphic function f on the unit disk partitions the unit circle, modulo a null set, into two disjoint pieces such that at each point of the first piece, f has a non-tangential limit, and at each point of the second piece, the cluster set of f in any Stolz angle is the entire plane. Higher dimensional versions of this result were first obtained by Calderon [2], who considered holomorphic functions on Cartesian products of half-planes. In this setting, an exact analogue of the one-dimensional result is obtained, in which the circle is replaced by the distinguished boundary, and the Stolz angles are replaced by products of cones in the coordinate half-planes. The ideas of Calderon were further developed by Rudin [8, pp. 79-83], who considered holomorphic and invariant harmonic functions in the ball of Cn. In this case, the circle is replaced by the unit sphere, and the Stolz angles are replaced by the approach regions of Korányi [4].

Boundary value conditions for wave fronts in reaction-diffusion systems

SynopsisIn this paper, we introduce a new class of solutions of reaction-diffusion systems, termed directional wave front solutions. They have a propagating character and the propagation direction selects some distinguished boundary points on which we can impose boundary conditions. The Neumann and Dirichlet problems on these points are treated here in order to prove some theorems on the existence of directional wave front solutions of small amplitude, and to partially establish their asymptotic behaviour.