An constructive proof for the Umemura polynomials for the third Painlev\'e equation

We are concerned with the Umemura polynomials associated with the third Painlev\'e equation. We extend Taneda's method, which was developed for the Yablonskii--Vorob'ev polynomials associated with the second Painlev\'e equation, to give an algebraic proof that the rational functions generated by the nonlinear recurrence relation satisfied by Umemura polynomials are indeed polynomials. Our proof is constructive and gives information about the roots of the Umemura polynomials.Comment: 20 pages, 3 figure

The effect of shared space on attitudes and behaviour

PhD ThesisShared space is a design approach that aims to reduce the dominance of motor vehicles by encouraging drivers to behave more accommodatingly towards pedestrians. The primary objectives of shared space in the UK are to improve pedestrian accessibility and safety. Despite its acknowledged advantages, research into the effects of shared space on vehicles and pedestrians remains limited. This research represents a unique opportunity to assess the impact of shared space by examining people's attitudes and behaviour before and after several shared space implementations. Factors that influence vehicle yielding and pedestrian gap acceptance behaviour were investigated at three case study sites in London and Bath. Predictive models for examining yielding and gap acceptance behaviour using logistic regression were developed. These models are a function of driver and vehicle attributes including gender and type of vehicle, pedestrian characteristics including assertiveness and age, and traffic conditions including the size of the gap between vehicles. The results suggest that drivers are statistically seven times more likely to yield to assertive pedestrians, and the presence of pedestrians in the roadway and the numbers of pedestrians waiting to cross both have a positive effect on yielding behaviour. Gap acceptance analysis indicated that pedestrians are statistically three times as likely to accept a gap between vehicles if there are other pedestrians already present in the roadway, and that female pedestrians are statistically 50% less likely to accept a gap than male pedestrians. Despite several statistically significant predictor variables, regression coefficients indicate that only around 30% of the variance in yielding behaviour and around 50% of the variance in gap acceptance behaviour can be explained by the models. This suggests that there are other unobservable variables that influence the behaviour of pedestrian and vehicles in shared space schemes. In order to further understand pedestrian and vehicle behaviour, road users were surveyed before and after a shared space implementation in Newcastle upon Tyne. Attitudes towards the shared space scheme were assessed through a survey of 500 road users before and 500 after the development. Attitudes were measured according to 29 items using an attitudinal scale. Statistically significant differences at the 95% level of confidence were recorded in how respondents perceive the accessibility of the environment, with improved perceptions after the development. Survey findings also show that perceptions of the negative IV consequences of vehicle traffic have been reduced after the development and the ambience of the street environment has improved. The angle and speed of crossing movements were recorded to analyse the effect of the development of the street on pedestrian crossing behaviour. There were some interesting findings, with pedestrians crossing on statistically significantly wider angles after the development, but no statistically significant differences in observations of pedestrian crossing speed. A sociability index was calculated to compare the number of people using the space in couples or groups to the number of people using the space on their own. No differences in sociability were found before and after the development, with an equal proportion of pedestrians observed using the street in couples or groups and on their own. Finally, to assess the effectiveness of the traffic calming measures, vehicle speed and volume were recorded before and after the development. There were statistically significant differences in vehicle speed, with mean average speeds decreasing by 13% after the shared space development and 85th percentile speeds decreasing by 19%. Vehicle volume also decreased after the development, with a reduction of 30%. The development was found to have been largely successful, with the introduction of shared space and reductions in vehicle flow and speed either having positive or neutral effects on pedestrian activity. This has positive implications for mixed-use local high streets, and suggests that shared space schemes can play a part in reversing the decline of such areas. In summary, this PhD research has demonstrated that shared spaces can help to nullify some of the negative effects of vehicle traffic, encourage cooperative behaviour between motorists and pedestrians and moderate the movement functions of mixed-use high streets

Classical Solutions of the Degenerate Fifth Painlev\'e Equation

In this paper classical solutions of the degenerate fifth Painlev\'e equation are classified, which include hierarchies of algebraic solutions and solutions expressible in terms of Bessel functions. Solutions of the degenerate fifth Painlev\'e equation are known to expressible in terms of the third Painlev\'e equation. Two applications of these classical solutions are discussed, deriving exact solutions of the complex sine-Gordon equation and of the coefficients in the three-term recurrence relation associated with generalised Charlier polynomials.Comment: 19 page