Anatomic foveal reconstruction of the triangular fibrocartilage complex with a tendon graft

An acute injury to the triangular fibrocartilage complex (TFCC) with avulsion of the foveal attachment can produce distal radioulnar joint (DRUJ) instability. The avulsed TFCC is translated distally so the footprint will be bathed in synovial fluid from the DRUJ and will become covered in synovitis. If the TFCC fails to heal to the footprint, then persistent instability can occur. The authors describe a surgical technique indicated for the treatment of persistent instability of the DRUJ due to foveal detachment of the TFCC. The procedure utilizes a loop of palmaris longus tendon graft passed through the ulnar aspect of the TFCC and into an osseous tunnel in the distal ulna to reconstruct the foveal attachment. This technique provides stability of the distal ulna to the radius and carpus. We recommend this procedure for chronic instability of the DRUJ due to TFCC avulsion, but recommend that suture repair remain the treatment of choice for acute instability. An arthroscopic assessment includes the trampoline test, hook test, and reverse hook test. DRUJ ballottement under arthroscopic vision details the direction of instability, the functional tear pattern, and unmasks concealed tears. If the reverse hook test demonstrates a functional instability between the TFCC and the radius, then a foveal reconstruction is contraindicated, and a reconstruction that stabilizes the radial and ulnar aspects of the TFCC is required. The foveal reconstruction technique has the advantage of providing a robust anatomically based reconstruction of the TFCC to the fovea, which stabilizes the DRUJ and the ulnocarpal sag.

The anisotropic λ -deformed SU(2) model is integrable

The all-loop anisotropic Thirring model interpolates between the WZW model and the non-Abelian T-dual of the anisotropic principal chiral model. We focus on the SU(2) case and we prove that it is classically integrable by providing its Lax pair formulation. We derive its underlying symmetry current algebra and use it to show that the Poisson brackets of the spatial part of the Lax pair, assume the Maillet form. In this way we procure the corresponding r and s matrices which provide non-trivial solutions to the modified Yang–Baxter equation.

Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality

We study representations of MV-algebras -- equivalently, unital lattice-ordered abelian groups -- through the lens of Stone-Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone-Priestley) dual spaces of MV-algebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MV-algebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone-Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0, 1]-valued functions on the spaces are isomorphic.