Direct computation of scattering matrices for general quantum graphs

We present a direct and simple method for the computation of the total scattering matrix of an arbitrary finite noncompact connected quantum graph given its metric structure and local scattering data at each vertex. The method is inspired by the formalism of Reflection-Transmission algebras and quantum field theory on graphs though the results hold independently of this formalism. It yields a simple and direct algebraic derivation of the formula for the total scattering and has a number of advantages compared to existing recursive methods. The case of loops (or tadpoles) is easily incorporated in our method. This provides an extension of recent similar results obtained in a completely different way in the context of abstract graph theory. It also allows us to discuss briefly the inverse scattering problem in the presence of loops using an explicit example to show that the solution is not unique in general. On top of being conceptually very easy, the computational advantage of the method is illustrated on two examples of "three-dimensional" graphs (tetrahedron and cube) for which other methods are rather heavy or even impractical.Comment: 20 pages, 4 figure

Factorization in integrable systems with impurity

This article is based on recent works done in collaboration with M. Mintchev, E. Ragoucy and P. Sorba. It aims at presenting the latest developments in the subject of factorization for integrable field theories with a reflecting and transmitting impurity

Symmetries of Spin Calogero Models

We investigate the symmetry algebras of integrable spin Calogero systems constructed from Dunkl operators associated to finite Coxeter groups. Based on two explicit examples, we show that the common view of associating one symmetry algebra to a given Coxeter group $W$ is wrong. More precisely, the symmetry algebra heavily depends on the representation of $W$ on the spins. We prove this by identifying two different symmetry algebras for a $B_L$ spin Calogero model and three for $G_2$ spin Calogero model. They are all related to the half-loop algebra and its twisted versions. Some of the result are extended to any finite Coxeter group.Comment: This is a contribution to the Special Issue on Dunkl Operators and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Exact results for the one-dimensional many-body problem with contact interaction: Including a tunable impurity

The one-dimensional problem of $N$ particles with contact interaction in the presence of a tunable transmitting and reflecting impurity is investigated along the lines of the coordinate Bethe ansatz. As a result, the system is shown to be exactly solvable by determining the eigenfunctions and the energy spectrum. The latter is given by the solutions of the Bethe ansatz equations which we establish for different boundary conditions in the presence of the impurity. These impurity Bethe equations contain as special cases well-known Bethe equations for systems on the half-line. We briefly study them on their own through the toy-examples of one and two particles. It turns out that the impurity can be tuned to lift degeneracies in the energies and can create bound states when it is sufficiently attractive. The example of an impurity sitting at the center of a box and breaking parity invariance shows that such an impurity can be used to confine asymmetrically a stationary state. This could have interesting applications in condensed matter physics.Comment: 20 pages, 5 figures, version accepted for publication: some typos corrected, references and comments adde

Quantum resolution of the nonlinear super-Schrodinger equation

We introduce a Z_2-graded version of the nonlinear Schrodinger equation that includes one fermion and one boson at the same time. This equation is shown to possess a supersymmetry which proves to be itself part of a super-Yangian symmetry based on gl(1|1). The solution exhibits a super version form of the classical Rosales solution. Then, we second quantize these results, and give a Lax pair formulation (based on gl(2|1)) for the model.Comment: 20 pages, no figur

Set-theoretical reflection equation: Classification of reflection maps

The set-theoretical reflection equation and its solutions, the reflection maps, recently introduced by two of the authors, is presented in general and then applied in the context of quadrirational Yang-Baxter maps. We provide a method for constructing reflection maps and we obtain a classification of solutions associated to all the families of quadrirational Yang-Baxter maps that have been classified recently

Factorization in integrable systems with impurity

This article is based on recent works done in collaboration with M. Mintchev, E. Ragoucy and P. Sorba. It aims at presenting the latest developments in the subject of factorization for integrable field theories with a reflecting and transmitting impurity.Comment: 7 pages; contribution to the XIVth International Colloquium on Integrable systems, Prague, June 200

Yang-Baxter and reflection maps from vector solitons with a boundary

Based on recent results obtained by the authors on the inverse scattering method of the vector nonlinear Schr\"odinger equation with integrable boundary conditions, we discuss the factorization of the interactions of N-soliton solutions on the half-line. Using dressing transformations combined with a mirror image technique, factorization of soliton-soliton and soliton-boundary interactions is proved. We discover a new object, which we call reflection map, that satisfies a set-theoretical reflection equation which we also introduce. Two classes of solutions for the reflection map are constructed. Finally, basic aspects of the theory of set-theoretical reflection equations are introduced.Comment: 29 pages. Featured article in Nonlinearit

The Syntax of British Sign Language: an Overview

The central aim of this research project is to identify and describe a range of grammatical structures found in British Sign Language, resulting in an account of the types of structures found and any possible motivations for their use. British Sign Language is the first or preferred language of a large number of Deaf people in Britain, and may have as many as 120,000 users (British Deaf Association, 2013). Despite Government recognition as an official language in March 2003 (United Kingdom Council on Deafness, 2003), there is little theoretical research, at an in-depth structural level, that can tell us much about the syntactic nature of the language. This research intends to expand the current knowledge of the syntactic processes occurring in British Sign Language, which has been established to some degree by Brennan et al., 1984; Kyle and Woll, 1985; Deuchar, 1984 and more recently Sutton-Spence and Woll, 1999, and Cormier, Smith and Sevcikova (2013, in press). With its central focus on clause structures, this research investigates the following questions: 1 What syntactic structures are found in BSL? 2 What are the frequencies of predicate types and clause structures? 3 What influences on syntax does the visual cognition of BSL users have? 4 What influences on syntax does the morphology of the language have? The analytical approach taken is to analyse British Sign Language entirely in its own terms and not to assume a priori a syntactic model derived from spoken languages. While conducting an inductive level of research with regards to the data, the approach is informed by cognitive linguistics (Croft and Cruse, 2009) and the semiogenetic model of signed languages (Fusellier-Souza, 2006; Slobin, 2008). The analysed data comprise samples of narratives selected from The British Sign Language Corpus, compiled by the Deafness Cognition and Language Research Centre, based at University College London. Presented in the form of quantitative tables, the analysis leads to statistics of types and frequencies of use of the central predicate structures found, as the study examines constituents within clauses and relationships that enable clause linkage. These types and frequencies are then considered in light of a cognitive explanation for their occurrence and illustrated by qualitative examples in boxes-within-boxes notation form (Kay, 2002)