A ring-closing metathesis route to 7-membered aza-heteroannulated sugars

Azepane rings have been constructed diastereoselectively upon a carbohydrate derivative utilising reductive amination and RCM. The stereochemistry of the ring junctions was confirmed by X-ray crystallography and NMR. Diastereoselective dihydroxylation has also been employed to afford a tetrahydroxylated azepane carbohydrate derivative with potential biological activity.     

The double duals and gauges of the Lanczos tensor

It is shown that: i) the Weyl tensor can be expressed in terms of the sum of a tensor and its double dual, where the tensor is constructed from the covariant derivatives of the Lanczos tensor, ii) a similar expression does not exist for the Riemann tensor in electromagnetic theory, iii) the electromagnetic field cannot be identified with the differential gauge freedom of the Lanczos tensor, iv) the symmetries of Einstein Maxwell theory and the Lanczos tensor do not prohibit the identification of the electromagnetic field with the algebraic gauge freedom of the Lanczos tensor, these symmetries require a differential equation relating the electromagnetic field tensor to the algebraic gauge vector and this is given.     

Resonant-cavity-enhanced photodetectors and LEDs in the mid-infrared

In this paper we outline the use of resonant-cavity enhancement for increasing the exterior coupling efficiency of photodetectors and light-emitting diodes (LEDs) in the mid-infrared (MIR) spectral region. This method is potentially very important in the MIR because encapsulation is not presently feasible due to the lack of suitable materials. Among other potential applications, resonant-cavity-enhanced (RCE) photodetectors and LEDs could be particularly suitable for greenhouse gas detection because of their ‘pre-tunable’ spectrally narrowed resonantly enhanced peaks. We also present the optical characterization of an InAs RCE photodetector aimed at the detection of methane gas (λ≈3.3 μm), and an InAs/InAs_0.91Sb_0.09 resonant-cavity LED (RCLED) aimed at carbon dioxide gas (λ≈4.2 μm). The high peak responsivity of the RCE photodetector was 34.7 A/W at λ=3.14 μm, and the RCLED peaked at λ=3.96 μm. These are among the longest operating wavelengths for III–V RCE photodetectors and RCLEDs reported in the literature.     

Galactic Metrics

The paths of stars in galaxies have circular velocity independent of their distance from the centre of the galaxy. Newtonian mechanics with a logarithmic potential has such paths. In relativity these paths can be taken to be geodesics and this requirement places a restricting equation on the spacetime metric. This equation has a non-unique spherically symmetric solution that in the Newtonian limit has a logarithmic potential. It also can be solved in terms of a conformal factor. In particular it has solutions which are conformal to the vacuum-Einstein solutions and such spacetimes are solutions to the vacuum-Bach equations. Therefore it is suggested that the Bach equations describe dynamics on galactic length scales.     

Thick grids with circular apertures: A comparison of theoretical and experimental performance

In this paper a comparison between a rigorous electromagnetic model for transmission through a hexagonal array of circular waveguides in a series of thick, metallic screens and experimental measurements in the far infrared is made. It is found that there is excellent agreement between theory and experiment when the frequency is below that where any diffracted orders propagate. The agreement is still very good above this frequency. Below a frequency approximately equal to the cut-off frequency of the circular waveguides little power is transmitted. As the thickness of the screen is increased, this decrease in transmission becomes more abrupt. Also, for thick screens, resonances appear in the transmission spectrum which are analogous to those which appear in the spectra of two grids separated by a distance comparable to the wavelength of the radiation being used.     

Resonant-induced transparency and coupled modes in layered metamaterials

We present an overview of some of our work on transmission properties and mode characteristics of layered metamaterials and their potential applications. In particular, we will show how layered metamaterials exhibit unusual transmission properties and unconventional features of guided modes, such as resonance-induced transparency in multilayer structures and slope reversal of dispersion curves in coupled waveguides. PACS 78.67.-n; 73.21.Ac; 61.46.+w; 42.25.Bs; 42.82.Et     

Ideals and solutions of nonlinear field equations

Families of horizontal ideals of contact manifolds of finite order are studied. Each horizontal ideal is shown to admit ann-dimensional module of Cauchy characteristic vectors that is also a module of annihilators (in the sense of Cartan) of the contact ideal. Since horizontal ideals are generated by 1-forms, any completely integrable horizontal ideal in the family leads to a foliation of the contact manifold by submanifolds of dimensionn on which the horizontal ideal vanishes. Explicit conditions are obtained under which an open subset of a leaf of this foliation is the graph of a solution map of the fundamental ideal that characterizes a given system of partial differential equations of finite order withn independent variables. The solution maps are obtained by sequential integration of systems of autonomous ordinary differential equations that are determined by the Cauchy characteristic vector fields for the problem. We show that every smooth solution map can be obtained in this manner. Let {_Vi¦1in} be a basis for the module of Cauchy characteristic vector fields that are in Jacobi normal form. If a subsidiary balance ideal admits each of then vector fieldsV _i as a smooth isovector field, then certain leaves of the foliation generated by the corresponding closed horizontal ideal are shown to be graphs of solution maps of the fundamental ideal. A subclass of these constructions agree with those of the Cartan-Kähler theorem. Conditions are also obtained under which every leaf of the foliation is the graph of a solution map. Solving a given system ofr partial differential equations withn independent variables on a first-order contact manifold is shown to be equivalent to the problem of constructing a complete system of independent first integrals. Properties of systems of first integrals are analyzed by studying the collection ISO[A _ij ] of all isovectors of the horizontal ideal. We show that ISO[A _ij ] admits the direct sum decomposition ^*[A _ij ]W[A _ij ] as a vector space, where ^*[A _ij ] is the module of Cauchy characteristics of the horizontal ideal. ISO[A _ij ] also forms a Lie algebra under the standard Lie product,^*[A _ij ] andW[A _ij ] are Lie subalgebras of ISO[A _ij ], and [A _ij ] is an ideal. A change of coordinates that resolves (straightens out) the canonical basis for ^*[A _ij ] is constructed. This change of coordinates is used to reduce the problem of solving the given system of PDE to the problem of root extraction of a system ofr functions ofn variables, and to establish the existence of solutions to a second-order system of overdetermined PDE that generate the subspaceW[A _ij ]. Similar results are obtained for second-order contact manifolds. Extended canonical transformations are studied. They are shown to provide algorithms for calculating large classes of closed horizontal ideals and a partial analog of classical Hamilton-Jacobi theory.