Boundary Integral Equations for the Laplace-Beltrami Operator

We present a boundary integral method, and an accompanying boundary element discretization, for solving boundary-value problems for the Laplace-Beltrami operator on the surface of the unit sphere $\S$ in $\mathbb{R}^3$. We consider a closed curve ${\cal C}$ on ${\cal S}$ which divides ${\cal S}$ into two parts ${\cal S}_1$ and ${\cal S}_2$. In particular, ${\cal C} = \partial {\cal S}_1$ is the boundary curve of ${\cal S}_1$. We are interested in solving a boundary value problem for the Laplace-Beltrami operator in $\S_2$, with boundary data prescribed on \C

On the multiple Borsuk numbers of sets

The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d. Our aim is to generalize this notion in the following way: The k-fold Borsuk number of such a set S is the smallest value of m such that there is a k-fold cover of S with m sets of diameters less than d. In this paper we characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give bounds for those of centrally symmetric sets, smooth bodies and convex bodies of constant width, and examine them for finite point sets in the Euclidean 3-space.Comment: 16 pages, 3 figure

The decline and rise of neighbourhoods: the importance of neighbourhood governance

There is a substantial literature on the explanation of neighbourhood change. Most of this literature concentrates on identifying factors and developments behind processes of decline. This paper reviews the literature, focusing on the identification of patterns of neighbourhood change, and argues that the concept of neighbourhood governance is a missing link in attempts to explain these patterns. Including neighbourhood governance in the explanations of neighbourhood change and decline will produce better explanatory models and, finally, a better view about what is actually steering neighbourhood change

Urban space and the social control of incivilities: perceptions of space influencing the regulationof anti-social behaviour

Contemporary cities are increasingly governed through space. In this article,we examine how urban space and perceptions thereof can influence the social control inthe area of incivilities. To this end, we first inspect the existing literature, in particularthe socio-spatial studies that emphasise the importance of culture and values in theinteraction with social control. Partly drawing on examples from our previous studies,we suggest that people’s perceptions of urban space (influenced by cultural symbols,social and media representations, aesthetics and other values) affect their perceptions ofincivilities, while the latter often determine or at least importantly contribute to theshaping of the social control of incivilities. We further highlight the role of gentrifica-tion as a medium and a tool of social control. The paper concludes by discussingimplications of this for the possible future, more integrated and interdisciplinaryresearch on the social control of incivilities in the city