A distributional approach to fragmentation equations

We consider a linear integro-di®erential equation that models multiple fragmentation with inherent mass-loss. A systematic procedure is presented for constructing a space of generalised functions Z0 in which initial-value problems involving singular initial conditions such as the Dirac delta distribution can be analysed. The procedure makes use of results on sun dual semigroups and quasi-equicontinuous semigroups on locally convex spaces. The existence and uniqueness of a distributional solution to an abstract version of the initial-value problem are established for any given initial data u0 in Z0

Open-street CCTV in Australia: the politics of resistance and expansion

This paper summarizes the first systematic attempt to document and assess the extent of open-street CCTV systems in Australia. In addition to providing empirical data, this paper argues that it is tempting for Australian scholars, and those elsewhere, to view the UK ‘surveillance revolution’ as the harbinger of inevitable global trends sweeping across jurisdictions. However analysis of the Australian data suggests that the deployment of CCTV in other national contexts may follow substantially divergent patterns. While the Australian CCTV experience follows many trends exhibited in other nations, it is nevertheless significant that the diffusion of CCTV in Australia has been more restrained than in the UK. We suggest that the divergence between the UK and Australian experiences resides in contrasting political structures and the consequent variation in the strength of debate and resistance at the local level

Three-dimensional coating and rimming flow : a ring of fluid on a rotating horizontal cylinder

The steady three-dimensional flow of a thin, slowly varying ring of Newtonian fluid on either the outside or the inside of a uniformly rotating large horizontal cylinder is investigated. Specifically, we study “full-ring” solutions, corresponding to a ring of continuous, finite and non-zero thickness that extends all the way around the cylinder. In particular, it is found that there is a critical solution corresponding to either a critical load above which no full-ring solution exists (if the rotation speed is prescribed) or a critical rotation speed below which no full-ring solution exists (if the load is prescribed). We describe the behaviour of the critical solution and, in particular, show that the critical flux, the critical load, the critical semi-width and the critical ring profile are all increasing functions of the rotation speed. In the limit of small rotation speed, the critical flux is small and the critical ring is narrow and thin, leading to a small critical load. In the limit of large rotation speed, the critical flux is large and the critical ring is wide on the upper half of the cylinder and thick on the lower half of the cylinder, leading to a large critical load. We also describe the behaviour of the non-critical full-ring solution, and, in particular, show that the semi-width and the ring profile are increasing functions of the load but, in general, non-monotonic functions of the rotation speed. In the limit of large rotation speed, the ring approaches a limiting non-uniform shape, whereas in the limit of small load, the ring is narrow and thin with a uniform parabolic profile. Finally, we show that, while for most values of the rotation speed and the load the azimuthal velocity is in the same direction as the rotation of the cylinder, there is a region of parameter space close to the critical solution for sufficiently small rotation speed in which backflow occurs in a small region on the upward-moving side of the cylinder

Fractional calculus of periodic distributions

Two approaches for defining fractional derivatives of periodic distributions are presented. The first is a distributional version of the Weyl fractional derivative in which a derivative of arbitrary order of a periodic distribution is defined via Fourier series. The second is based on the Gr¨unwald-Letnikov formula for defining a fractional derivative as a limit of a fractional difference quotient. The equivalence of the two approaches is established and an application to a fractional diffusion equation, posed in a space of periodic distributions, is also discuss

The Dynamics of School Attainment of Englands Ethnic Minorities

We exploit a universe dataset of state school students in England with linked test score records to document the evolution of attainment through school for different ethnic groups. The analysis yields a number of striking findings. First, we show that, controlling for personal characteristics, all minority groups make greater progress than white students over secondary schooling. Second, much of this improvement occurs in the high-stakes exams at the end of compulsory schooling. Third, we show that for most ethnic groups, this gain is pervasive, happening in almost all schools in which these students are found. We address some of the usual factors invoked to explain attainment gaps: poverty, language, school quality, and teacher influence. We conclude that our findings are more consistent with the importance of factors like aspirations and attitudes.Ethnic test score gap, school attainment, education

Fractional transformations of generalised functions

A distributional theory of fractional transformations is developed. A constructive approach, based on the eigenfunction expansion method pioneered by A. H. Zemanian, is used to produce an appropriate space of test functions and corresponding space of generalised functions. The fractional transformations that are defined are shown to form an equicontinuous group of operators on the space of test functions and a weak continuous group on the space of generalised functions. Integral representations for the fractional transformations are also obtained under certain conditions. The fractional Fourier transformation is considered as a particular case of our general theory