Uncovering Fractional Monodromy

The uncovering of the role of monodromy in integrable Hamiltonian fibrations has been one of the major advances in the study of integrable Hamiltonian systems in the past few decades: on one hand monodromy turned out to be the most fundamental obstruction to the existence of global action-angle coordinates while, on the other hand, it provided the correct classical analogue for the interpretation of the structure of quantum joint spectra. Fractional monodromy is a generalization of the concept of monodromy: instead of restricting our attention to the toric part of the fibration we extend our scope to also consider singular fibres. In this paper we analyze fractional monodromy for n ₁:(?n ₂) resonant Hamiltonian systems with n ₁, n ₂ coprime natural numbers. We consider, in particular, systems that for n ₁, n ₂ > 1 contain one-parameter families of singular fibres which are ‘curled tori’. We simplify the geometry of the fibration by passing to an appropriate branched covering. In the branched covering the curled tori and their neighborhood become untwisted thus simplifying the geometry of the fibration: we essentially obtain the same type of generalized monodromy independently of n ₁, n ₂. Fractional monodromy is then recovered by pushing the results obtained in the branched covering back to the original system.     

Optical activity of chiral thin film and liquid crystal hybrids

Using the glancing angle deposition (GLAD) technique, we have fabricated porous, chiral thin films with optically anisotropic helical microstructures that exhibit optical phenomena such as wavelength specific rotation of linearly polarized light. Initial research has shown that the porosity of the films allows for the addition of nematic liquid crystals (NLCs) to the films for promising applications in dynamically switchable devices, while simultaneously enhancing the optical properties of the film. This study describes the fundamental optical behaviour of LC-filled chiral thin films in relation to material, porosity, structure and thickness. It was found that for SiO₂ films, the addition of NLCs increased the effective rotatory power by two-fold when compared with results from the film without added LCs. The rotatory power of Al₂O₃ and MgF₂ films, while being similarly increased by the addition of LCs, exhibited a reversal in sign, or direction of rotation, for the visible wavelength spectrum investigated. The effects of film porosity and structure were studied by varying the angle of incidence from 83° to 86°; it was found that the greater porosity of the films deposited at larger angles allowed for more filling by the LCs and thus a larger increase in rotatory power. Finally, the addition of LCs was observed to shift the wavelength of peak rotation towards smaller values.     

A dielectric study on the relaxation and switching behaviour of liquid crystals confined within a colloidal network

The molecular ordering and dynamics of a liquid crystal (LC E7) in the presence of a three-dimensional network of submicron particles have been studied by dielectric relaxation spectroscopy. The field-dependent orientation of the LC was quantified by the director order parameter and modelled by use of a three-phase model. The influence of the colloidal network on the molecular dynamics was assessed from the dielectric spectra, e.g. from the position of relaxation peaks as well as from the strength of the two principal relaxations (α and γ). The spectra changed noticeably upon application of an increasing d.c. bias. A reduction of the threshold field was observed upon addition of colloidal particles to the LC. This was associated with a switching between two metastable states induced by anchoring on the filler particles. Modelled spectra were found to be in good agreement with the experimental data. The modelling showed that the confined LC phase is composed of two fractions, viz. an ordered and a disordered one with different molecular mobilities. Furthermore, switching experiments were conducted at various temperatures in order to evaluate the impact of the colloidal network on the (temperature-dependent) orientational behaviour of the LC molecules. For the colloid-filled LC higher conductivities were found, which gave rise to longer switch-off times.     

Dynamics and geometry near resonant bifurcations

This paper provides an overview of the universal study of families of dynamical systems undergoing a Hopf-Ne?marck-Sacker bifurcation as developed in [1–4]. The focus is on the local resonance set, i.e., regions in parameter space for which periodic dynamics occurs. A classification of the corresponding geometry is obtained by applying Poincaré-Takens reduction, Lyapunov-Schmidt reduction and contact-equivalence singularity theory, equivariant under an appropriate cyclic group. It is a classical result that the local geometry of these sets in the nondegenerate case is given by an Arnol’d resonance tongue. In a mildly degenerate situation a more complicated geometry given by a singular perturbation of a Whitney umbrella is encountered. Our approach also provides a skeleton for the local resonant Hopf-Ne?marck-Sacker dynamics in the form of planar Poincaré-Takens vector fields. To illustrate our methods a leading example is used: A periodically forced generalized Duffing-Van der Pol oscillator.     

Flux line lattice symmetries in the borocarbide superconductor LuNi2B2C

We compare the results of small angle neutron scattering on the flux line lattice (FLL) obtained in the borocarbide superconductor LuNi₂B₂C with the applied field along the c- and a-axes. For H‖c the temperature dependence of the FLL structural phase transition from square to hexagonal symmetry was investigated. Above 10 K the transition onset field. H ₂(T), rises sharply, bending away from H _c2(T) in contradiction to theoretical predictions of the two merging. For H‖a a first order FLL reorientation transition is observed at H _tr=3–3.5 kOe. Below H _tr the FLL nearest neighbor direction is parallel to the b-axis, and above H _tr to the c-axis. This transition cannot be explained using nonlocal corrections to the London model.     

The Direct Summand Property in Modular Invariant Theory

In nonmodular invariant theory of finite groups, the invariant ring is always a direct summand of the full polynomial ring. This is no longer generally true in modular invariant theory, but nevertheless interesting examples are known where this happens. We give useful characterisations of the direct summand property in terms of the image of a twisted transfer map. For example, for p-groups acting in characteristic p the direct summand property holds if and only if the image of the ordinary transfer is a principal ideal; and in that case the group is generated by transvections. We also extend some known results in the nonmodular case to where only the direct summand property is assumed, e.g., the invariant ring is always generated by its elements of degree at most the order of the group.     

Unfoldings of quasi-periodic tori in reversible systems

A general KAM-theory for reversible systems is given. The cases of both maximal and lower-dimensional tori are covered. In some cases parameters are needed for persistence, therefore an unfolding theory is developed.     

Some properties of the solutions of wave equations

Some properties of solutions of initial value problems and mixed initial-boundary value problems of a class of wave equations are discussed. Wave modes are defined and it is shown that for the given class of wave equations there is a one to one correspondence with the roots _i (k) or k _j () of the dispersion relation W(, k)=0. It is shown that solutions of initial value problems cannot consist of single wave modes if the initial values belong to W ₂ ¹ (–, ); generally such solutions must contain all possible modes. Similar results hold for solutions of mixed initial-boundary value problems. It is found that such solutions are stable, even if some of the singularities of the functions k _j () lie in the upper half of the plane. The implications of this result for the Kramers-Kronig relations are discussed.     

Control over the morphology of porous polymeric membranes for flow through biosensors

Porous membranes have been developed by photo-polymerisation-induced phase separation of a monomer/solvent mixture. Characterised by a large open surface area reaching up to about 50 m²/g, these membranes are suitable for flow through biosensor applications in the field of molecular diagnostics. The effects of the polymerisation conditions and the phase separation on the resulting structures are discussed. SEM analysis and BET surface area measurements revealed that morphology can be well controlled by adjusting monomer concentration and UV intensity. Membranes functionalised with epoxy and amine groups show high coupling efficiency of oligo-DNA and large hybridisation efficiency.