An exactly solvable four transition point model

Heavy particle collisions, in particular low-energy ion-atom collisions, are amenable to semiclassical JWKB phase integral analysis in the complex plane of the internuclear separation. Analytic continuation in this plane requires due attention to the Stokes phenomenon which parametrizes the physical mechanisms of curve crossing, non-crossing, the hybrid Nikitin model, rotational coupling and predissociation. Complex transition points represent adiabatic degeneracies. In the case of two or more such points, the Stokes constants may only be completely determined by resort to the so-called comparison- equation method involving, in particular, parabolic cylinder functions or Whittaker functions and their strong-coupling asymptotics. In particular, the Nikitin model is a two transition-point one-double-pole problem in each half-plane corresponding to either ingoing or outgoing waves. When the four transition points are closely clustered, new techniques are required to determine Stokes constants. However, such investigations remain incomplete, A model problem is therefore solved exactly for scattering along a one-dimensional z-axis. The energy eigenvalue is b(2)-a(2) and the potential comprises -z(2)/2 (parabolic) and -a(2) + b(2)/2z(2) (centrifugal/centripetal) components. The square of the wavenumber has in the complex z-plane, four zeros each a transition point at z = +/-a +/- ib and has a double pole at z = 0. In cases (a) and (b), a and b are real and unitarity obtains. In case (a) the reflection and transition coefficients are parametrized by exponentials when a(2) + b(2) > 1/2. In case (b) they are parametrized by trigonometrics when a(2) + b(2) <1/2 and total reflection is achievable. In case (c) a and b are complex and in general unitarity is not achieved due to loss of flux to a continuum (O'Rourke and Crothers, 1992 Proc. R. Sec. 438 1). Nevertheless, case (c) coefficients reduce to (a) or (b) under appropriate limiting conditions. Setting z = ht, with h a real constant, an attempt is made to model a two-state collision problem modelled by a pair of coupled first-order impact parameter equations and an appropriate (T) over tilde-tau relation, where (T) over tilde is the Stueckelberg variable and tau is the reduced or scaled time. The attempt fails because (T) over tilde is an odd function of tau, which is unphysical in a real collision problem. However, it is pointed out that by applying the Kummer exponential model to each half-plane (O'Rourke and Crothers 1994 J. Phys. B: At. Mel. Opt. Phys. 27 2497) the current model is in effect extended to a collision problem with four transition points and a double pole in each half-plane. Moreover, the attempt in itself is not a complete failure since it is shown that the result is a perfect diabatic inelastic collision for a traceless Hamiltonian matrix, or at least when both diagonal elements are odd and the off-diagonal elements equal and even.

A New Restricted Full-Rank Single-Symbol Decodable Design for Four Transmit Antennas

Recently, a single-symbol decodable transmit strategy based on preprocessing at the transmitter has been introduced to decouple the quasi-orthogonal space-time block codes (QOSTBC) with reduced complexity at the receiver [9]. Unfortunately, it does not achieve full diversity, thus suffering from significant performance loss. To tackle this problem, we propose a full diversity scheme with four transmit antennas in this letter. The proposed code is based on a class of restricted full-rank single-symbol decodable design (RFSDD) and has many similar characteristics as the coordinate interleaved orthogonal designs (CIODs), but with a lower peak-to-average ratio (PAR).

Emotional development among early school-age children: Gender differences in the role of problem behaviours

There has been an increasing focus on social and emotional development in educational programmes in early childhood as both variables are believed to influence behavioural outcomes in the classroom. However, relationships between social and emotional development and behaviour in early childhood have rarely been explored. This article sets out to investigate the conceptualisation of these variables and their inter-relationships. Structural equation models were used to assess if differences exist between boys and girls in relation to social and emotional competences, which could affect the relative success of such programmes. This article is based on cross-sectional data collected from 749 four- to six-year-olds and their teachers. The findings generally supported the hypothesised relationships between social and emotional development variables and prosocial behaviour (including internalising behaviour) for boys and girls. However, some gender differences were noted in externalising behaviour, which teachers often consider to be most significant due to its potentially disruptive nature in the classroom.