Rigid motions: action-angles, relative cohomology and polynomials with roots on the unit circle

Revisiting canonical integration of the classical solid near a uniform rotation, canonical action angle coordinates, hyperbolic and elliptic, are constructed in terms of various power series with coefficients which are polynomials in a variable $r^2$ depending on the inertia moments. Normal forms are derived via the analysis of a relative cohomology problem and shown to be obtainable without the use of ellitptic integrals (unlike the derivation of the action-angles). Results and conjectures also emerge about the properties of the above polynomials and the location of their roots. In particular a class of polynomials with all roots on the unit circle arises.Comment: 26 pages, 1 figur

Partition function of the eight-vertex model with domain wall boundary condition

We derive the recursive relations of the partition function for the eight-vertex model on an $N\times N$ square lattice with domain wall boundary condition. Solving the recursive relations, we obtain the explicit expression of the domain wall partition function of the model. In the trigonometric/rational limit, our results recover the corresponding ones for the six-vertex model.Comment: Latex file, 20 pages; V2, references adde

On a generalization of Jacobi's elliptic functions and the Double Sine-Gordon kink chain

A generalization of Jacobi's elliptic functions is introduced as inversions of hyperelliptic integrals. We discuss the special properties of these functions, present addition theorems and give a list of indefinite integrals. As a physical application we show that periodic kink solutions (kink chains) of the double sine-Gordon model can be described in a canonical form in terms of generalized Jacobi functions.Comment: 18 pages, 9 figures, 3 table

The eVALuate study: two parallel randomised trials, one comparing laparoscopic with abdominal hysterectomy, the other comparing laparoscopic with vaginal hysterectomy

OBJECTIVE: To compare the effects of laparoscopic hysterectomy and abdominal hysterectomy in the abdominal trial, and laparoscopic hysterectomy and vaginal hysterectomy in the vaginal trial. DESIGN: Two parallel, multicentre, randomised trials. Setting 28 UK centres and two South African centres. Participants 1380 women were recruited; 1346 had surgery; 937 were followed up at one year. PRIMARY OUTCOME: outcome Rate of major complications. RESULTS: In the abdominal trial laparoscopic hysterectomy was associated with a higher rate of major complications than abdominal hysterectomy (11.1% v 6.2%, P = 0.02; difference 4.9%, 95% confidence interval 0.9% to 9.1%) and the number needed to treat to harm was 20. Laparoscopic hysterectomy also took longer to perform (84 minutes v 50 minutes) but was less painful (visual analogue scale 3.51 v 3.88, P = 0.01) and resulted in a shorter stay in hospital after the operation (3 days v 4 days). Six weeks after the operation, laparoscopic hysterectomy was associated with less pain and better quality of life than abdominal hysterectomy (SF-12, body image scale, and sexual activity questionnaires). In the vaginal trial we found no evidence of a difference in major complication rates between laparoscopic hysterectomy and vaginal hysterectomy (9.8% v 9.5%, P = 0.92; difference 0.3%, − 5.2% to 5.8%), and the number needed to treat to harm was 333.We found no evidence of other differences between laparoscopic hysterectomy and vaginal hysterectomy except that laparoscopic hysterectomy took longer to perform (72 minutes v 39 minutes) and was associated with a higher rate of detecting unexpected pathology (16.4% v 4.8%, P = < 0.01). However, this trial was underpowered. CONCLUSIONS: Laparoscopic hysterectomy was associated with a significantly higher rate of major complications than abdominal hysterectomy. It also took longer to perform but was associated with less pain, quicker recovery, and better short term quality of life. The trial comparing vaginal hysterectomy with laparoscopic hysterectomy was underpowered and is inconclusive on the rate of major complications; however, vaginal hysterectomy took less time

Particles in a magnetic field and plasma analogies: doubly periodic boundary conditions

The $N$-particle free fermion state for quantum particles in the plane subject to a perpendicular magnetic field, and with doubly periodic boundary conditions, is written in a product form. The absolute value of this is used to formulate an exactly solvable one-component plasma model, and further motivates the formulation of an exactly solvable two-species Coulomb gas. The large $N$ expansion of the free energy of both these models exhibits the same O(1) term. On the basis of a relationship to the Gaussian free field, this term is predicted to be universal for conductive Coulomb systems in doubly periodic boundary conditions.Comment: 12 page

Spacetime Encodings II - Pictures of Integrability

I visually explore the features of geodesic orbits in arbitrary stationary axisymmetric vacuum (SAV) spacetimes that are constructed from a complex Ernst potential. Some of the geometric features of integrable and chaotic orbits are highlighted. The geodesic problem for these SAV spacetimes is rewritten as a two degree of freedom problem and the connection between current ideas in dynamical systems and the study of two manifolds sought. The relationship between the Hamilton-Jacobi equations, canonical transformations, constants of motion and Killing tensors are commented on. Wherever possible I illustrate the concepts by means of examples from general relativity. This investigation is designed to build the readers' intuition about how integrability arises, and to summarize some of the known facts about two degree of freedom systems. Evidence is given, in the form of orbit-crossing structure, that geodesics in SAV spacetimes might admit, a fourth constant of motion that is quartic in momentum (by contrast with Kerr spacetime, where Carter's fourth constant is quadratic).Comment: 11 pages, 10 figure

A slowly rotating perfect fluid body in an ambient vacuum

A global model of a slowly rotating perfect fluid ball in general relativity is presented. To second order in the rotation parameter, the junction surface is an ellipsoidal cylinder. The interior is given by a limiting case of the Wahlquist solution, and the vacuum region is not asymptotically flat. The impossibility of joining an asymptotically flat vacuum region has been shown in a preceding work.Comment: 7 pages, published versio

Free Energy of the Eight Vertex Model with an Odd Number of Lattice Sites

We calculate the bulk contribution for the doubly degenerated largest eigenvalue of the transfer matrix of the eight vertex model with an odd number of lattice sites N in the disordered regime using the generic equation for roots proposed by Fabricius and McCoy. We show as expected that in the thermodynamic limit the result coincides with the one in the N even case.Comment: 11 pages LaTeX New introduction, Method change

Generating Complex Potentials with Real Eigenvalues in Supersymmetric Quantum Mechanics

In the framework of SUSYQM extended to deal with non-Hermitian Hamiltonians, we analyze three sets of complex potentials with real spectra, recently derived by a potential algebraic approach based upon the complex Lie algebra sl(2, C). This extends to the complex domain the well-known relationship between SUSYQM and potential algebras for Hermitian Hamiltonians, resulting from their common link with the factorization method and Darboux transformations. In the same framework, we also generate for the first time a pair of elliptic partner potentials of Weierstrass $\wp$ type, one of them being real and the other imaginary and PT symmetric. The latter turns out to be quasiexactly solvable with one known eigenvalue corresponding to a bound state. When the Weierstrass function degenerates to a hyperbolic one, the imaginary potential becomes PT non-symmetric and its known eigenvalue corresponds to an unbound state.Comment: 20 pages, Latex 2e + amssym + graphics, 2 figures, accepted in Int. J. Mod. Phys.

Dynamic Fano Resonance of Quasienergy Excitons in Superlattices

The dynamic Fano resonance (DFR) between discrete quasienergy excitons and sidebands of their ionization continua is predicted and investigated in dc- and ac-driven semiconductor superlattices. This DFR, well controlled by the ac field, delocalizes the excitons and opens an intrinsic decay channel in nonlinear four-wave mixing signals.Comment: 4pages, 4figure