A problem posed in the early eighteenth century asks for right-angled triangles, each of whose sides exceeds double the area by a perfect square. We summarize known results and find such triangles with the smallest possible standard generators.

This article sets out to examine ethnic differences in the key influences on quality of life for older people in the context of the increasing health and wealth of British older people generally and the ageing of the post-1945 migrants. It is based on secondary multivariate analysis of the Fourth National Survey of Ethnic Minorities of England and Wales. Respondents aged 45–74 years belonging to four ethnic groups (1,068 white, 514 Caribbean, 581 Indian and East African Asian, and 199 Pakistani) were included in the analysis, which focuses on differences between ethnic groups by age and gender, using the white population as the reference group. Four dimensions (incorporating seven factors) that influence the quality of life were determined among this age group: quality of neighbourhood (availability of local amenities, and problems with crime and the physical environment); social networks and community participation (strength of family networks, and community participation); material conditions (income, wealth and housing conditions) and health. The relative position of the four ethnic groups on the seven factors illustrated two contrasting patterns. For the factors based on conventional indicators of social inequalities – such as material circumstances, health, participation in formal social networks, and quality of the physical environment – the white group ranked highest, the Pakistanis lowest, and the Indian and Caribbean groups ranked second and third. But factors that capture more immediate and subjective elements, such as frequency of family contact and the desirability of the residential neighbourhood, displayed a diametrically opposite rank-order, with the Pakistani group ranked first and the white group fourth. The study highlights the value of examining separately the various influences on quality of life. Contradictory patterns are revealed in key influences that are hidden by global measures. The study also reveals the difficulty of identifying culturally-neutral measures of quality of locality, with ethnic minority groups having a more positive perception of their area than rated by conventional measures of area deprivation such as the Index of Deprivation.

This paper is devoted to a study of the integral of the workload process of the single server queue, in particular during one busy period. Firstly, we find asymptotics of the area 𝒜 swept under the workload process W(t) during the busy period when the service time distribution has a regularly varying tail. We also investigate the case of a light-tailed service time distribution. Secondly, we consider the problem of obtaining an explicit expression for the distribution of 𝒜. In the general GI/G/1 case, we use a sequential approximation to find the Laplace—Stieltjes transform of 𝒜. In the M/M/1 case, this transform is obtained explicitly in terms of Whittaker functions. Thirdly, we consider moments of 𝒜 in the GI/G/1 queue. Finally, we show asymptotic normality of .

This paper is concerned with the distibutions and the moments of the area and the local time of a random walk, called the Bernoulli meander. The limit behavior of the distributions and the moments is determined in the case where the number of steps in the random walk tends to infinity. The results of this paper yield explicit formulas for the distributions and the moments of the area and the local time for the Brownian meander.

Kac's formula for Brownian functionals and Levy's local time decomposition are shown to be useful tools in analysing Brownian excursion properties. These tools are applied to maximum, local time and area distributions. Some curious connections between some of these distributions are explained by simple probabilistic arguments.

Some properties of the Voronoi polygon associated with a moving particle in a regular tessellation are studied. In particular, the area of the variable polygon has played a relevant role in previous studies on plant competition, animal predation, adjustment of the sex ratio in bisexual populations and the gregarious behaviour of animal species.