On uncertainty bounds and growth estimates for fractional fourier transforms

Gelfand–Shilov spaces are spaces of entire functions defined in terms of a rate of growth in one direction and a concomitant rate of decay in an orthogonal direction. Gelfand and Shilov proved that the Fourier transform is an isomorphism among certain of these spaces. In this article we consider mapping properties of fractional Fourier transforms on Gelfand–Shilov spaces. Just as the Fourier transform corresponds to rotation through a right angle in the phase plane, fractional Fourier transforms correspond to rotations through intermediate angles. Therefore, the aim of fractional Fourier estimates is to set up a correspondence between growth properties in the complex plane versus decay properties in phase space.     

The Fractional Clifford-Fourier Kernel

The Clifford-Fourier transform was introduced by Brackx, De Schepper and Sommen who subsequently computed its kernel in dimension d=2. Here we compute the kernel of a fractional version of the transform when d=2 and 4. In doing so we solve appropriate wave-type problems on spheres in two and four dimensions. We also give formulae for the solutions of these problems in all even dimensions and hence a means of computing the fractional Clifford-Fourier kernels in even dimensions.     

Emission from the working and counter electrodes under co-reactant electrochemiluminescence conditions

We present a new approach to explore the potential-dependent multi-colour co-reactant electrochemiluminescence (ECL) from multiple luminophores. The potentials at both the working and counter electrodes, the current between these electrodes, and the emission over cyclic voltammetric scans were simultaneously measured for the ECL reaction of Ir(ppy)₃ and either [Ru(bpy)₃]²⁺ or [Ir(df-ppy)₂(ptb)]⁺, with tri-n-propylamine as the co-reactant. The counter electrode potential was monitored by adding a differential electrometer module to the potentiostat. Plotting the data against the applied working electrode potential and against time provided complementary depictions of their relationships. Photographs of the ECL at the surface of the two electrodes were taken to confirm the source of the emissions. This provided a new understanding of these multifaceted ECL systems, including the nature of the counter electrode potential and the possibility of eliciting ECL at this electrode, a mechanism-based rationalisation of the interactions of different metal-complex luminophores, and a previously unknown ECL pathway for the Ir(ppy)₃ complex at negative potentials that was observed even in the absence of the co-reactant.

Exploration of potential-dependent, multi-colour co-reactant electrochemiluminescence from multiple luminophores at the working and counter electrodes reveals new pathways to emission.     

Direct observation of the phenomenology of a solid thermal explosion using time-resolved proton radiography

We present a new phenomenology for burn propagation inside a thermal explosion based on dynamic radiography. Radiographic images were obtained of an aluminum cased solid cylindrical sample of a plastic bonded formulation of octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine. The phenomenology observed is ignition followed by cracking in the solid accompanied by the propagation of a radially symmetric front of increasing proton transmission. This is followed by a further increase in transmission through the sample, ending after approximately 100 micros. We show that these processes are consistent with the propagation of a convective burn front followed by consumption of the remaining solid by conductive particle burning.     

Synthetic applications of carbolithiation transformations

Carbolithiation reactions are exceptionally versatile transformations which have been utilised in a remarkably diverse and creative manner. In this review we outline the background and scope of these reactions and then focus on their use in organic synthesis with a particular emphasis on literature examples published since 2000.     

Asymmetric cascade reaction sequences via chiral lithiated intermediates

The (-)-sparteine-mediated enantioselective intermolecular carbolithiation of (E)-2-propenylarylamines allows for the generation of chiral lithiated intermediates which have broad synthetic potential. These intermediates have been exploited in a series of further in situ reactions with electrophiles to generate a collection of products each containing a common stereogenic center. The stereogenic center, formed in high enantiomeric ratio in the first carbolithiation step, is carried through the cascade reaction sequence to the final products and is independent of electrophile used. The methodology is demonstrated by the synthesis of structurally diverse chiral anilines, indoles, and indolones all with an er of 92:8 (+/-1). The heterocyclic syntheses involve an enantioselective alkene carbolithiation and subsequent trapping of the intermediate organolithium with a suitable electrophile, followed by an in situ ring closure and dehydration to generate the indole or indolone rings.     

Enantioselective carbolithiation initiated cascade reactions

Enantioselective cascade reactions are very powerful synthetic protocols for the assembly of complex architectures. Our current approach is to exploit a (-)-sparteine-controlled enantioselective carbolithiation of 2-propenylarylamines to provide chiral intermediates, which depending upon choice of electrophile are subsequently in situ converted into products of increasing complexity. The chiral center, formed in high enantiomeric ratio in the first carbolithiation step, is carried through the cascade reaction sequence to the final products and is independent of electrophile used. The scope of the cascade reaction is outlined in terms of reaction conditions, alkyllithiums, and electrophiles.     

Testing ion-neutral interaction potentials using calculated ion transport coefficients

Several commonly measured ion transport coefficients were investigated in order to determine their sensitivity for testing and comparing proposed ion-neutral interaction potentials. A variety of positive ions, negative ions, neutrals, and temperatures were included in order to draw as general a conclusion as possible. All transport coefficients considered were found to be sufficiently sensitive to be used to clearly distinguish between less and more accurate interaction potentials. It was also found that the longitudinal diffusion coefficient is the most sensitive test, followed by both the transverse diffusion coefficient and the ratio of the longitudinal diffusion coefficient to mobility, followed by the ratio of the transverse diffusion coefficient to mobility and that the mobility is the least sensitive test. When presently achievable levels of experimental error were also taken into account, however, there was no significant difference in the sensitivities.