Singular Dirichlet boundary value problems II: Resonance case

Existence results are established for the resonant problem y + _m a y = f(t, y) a.e. on [0, 1] with y satisfying Dirichlet boundary conditions. The problem is singular since f is a Carathéodory function, with a > 0 a.e. on [0, 1] and      

On periodic solutions for even order differential equations

A class of sufficient conditions are obtained for the existence of a unique 2π$2\pi$-periodic solution of even order differential equations, employing an initial value problem.     

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.     

On abstract degenerate neutral differential equations

We introduce a new abstract model of functional differential equations, which we call abstract degenerate neutral differential equations, and we study the existence of strict solutions. The class of problems and the technical approach introduced in this paper allow us to generalize and extend recent results on abstract neutral differential equations. Some examples on nonlinear partial neutral differential equations are presented.     

Integrated thin-film polymer/fullerene photodetectors for on-chip microfluidic chemiluminescence detection

We report the use of solution-processed thin-film organic photodiodes for microscale chemiluminescence. The active layer of the photodiodes comprised a 1 : 1 blend by weight of the conjugated polymer poly(3-hexylthiophene) [P3HT] and [6,6]-phenyl-C(61)-butyric acid-methylester [PCBM]--a soluble derivative of C(60). The devices had an active area of 1 mm x 1 mm, and a broad-band response from 350 to 700 nm, with an external quantum efficiency of more than 50% between 450 and 550 nm. The photodiodes have a simple layered structure that permits facile integration with planar chip-based systems. To evaluate the suitability of the organic devices as integrated detectors for microscale chemiluminescence, a peroxyoxalate based chemiluminescence reaction (PO-CL) was monitored within a poly(dimethyl-siloxane) (PDMS) microfluidic device. Quantitation of hydrogen peroxide indicated excellent linearity and yielded a detection limit of 10 microM, comparable with previously reported results using micromachined silicon microfluidic chips with integrated silicon photodiodes. The combination of organic photodiodes with PDMS microfluidic chips offers a means of creating compact, sensitive and potentially low-cost microscale CL devices with wide-ranging applications in chemical and biological analysis and clinical diagnostics.     

Oscillation of second-order damped dynamic equations on time scales

The study of dynamic equations on time scales has been created in order to unify the study of differential and difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which may be an arbitrary closed subset of the reals. This way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained. In this paper, by employing the Riccati transformation technique we will establish some oscillation criteria for second-order linear and nonlinear dynamic equations with damping terms on a time scale . Our results in the special case when and extend and improve some well-known oscillation results for second-order linear and nonlinear differential and difference equations and are essentially new on the time scales , h>0, for q>1, , etc. Some examples are considered to illustrate our main results.