Environment and Planning B: Urban Analytics and City Science

This paper describes the economical placing of braces in the walls and ceiling of a rectangular one-story building. First, efficient schemes for bracing the ceiling are shown to correspond to trees in a bipartite graph. Then a combinatorial analysis shows how certain sets of braces in vertical walls cause some ceiling braces to be redundant. We also give a complete description of bracing schemes for plane grids of squares.

The fundamental tools are elementary vector space theory and combinatorial geometry (matroid theory). However, our style is deliberately discursive, with many worked examples, because we hope to attract readers possessing mathematical sophistication and mechanical intuition at a wide range of levels.

The paper describes the minimally redundant sets of diagonal braces in a grid of cubes in space. The language developed is essentially graphical. The methods extend those of the first few sections of an earlier work by Bolker and Crapo (1977).

The paper discusses the improvement of the thermal insulation of existing houses to reduce their energy consumption, with particular emphasis on solid walled buildings, and gives details of some of the methods available for insulating such buildings. The techniques used to improve the insulation of a Victorian house in Cambridgeshire are described in detail and the U values and heat loss of the improved house are compared with the original building and with current Building Regulations. The reduced energy demand allows the use of an unconventional heating system using coke and wood as fuels, and relying on natural convection to distribute the heat throughout the house. The operation of the house is described together with estimates of fuel consumption, which have been confirmed by the first year of operation.

This paper presents a comparison of five methods of representing a general spatial rigid-body rotation about a fixed point. The following representations are considered: the real orthogonal 3 × 3 matrix; the special unitary 2 × 2 matrix; the Pauli spin matrices; the unit quaternion; and the special unitary 3 × 3 matrix together with spherical harmonics of the first degree. Although the first of these representations is certainly the most commonly used, particularly in engineering and technological applications, it is shown that it is not the best or most efficient representation. The conclusion reached is that the most concise and efficient representation in practice is the unit quaternion, although the special unitary 2 × 2 matrices follow closely behind.

In this paper the urban-planning process is explored and modelled using a variety of concepts and techniques drawn from the theory of games, The rationale for using game theory as a basis for simulating the design process is presented first, and this serves to highlight the major features of such processes in terms of bargaining and the implied power positions of the players involved. In the second section these ideas are given substance through a description of a case study based on the choice of location of a town to accept overspill population from a large conurbation, and a number of conceptual game-theoretic models of parts of this process are presented. By developing game theory nonalgebraically in terms of this case study, it is then possible to generate a set of formal models based on stochastic game thoery, as first suggested by Shapley (1953). These models are presented theoretically in the third section, and in terms of their algorithms and application in section four. These models include several different features including a multigame stochastic format in which participants move between game elements according to transition probabilities conditional on their joint decisions, an hierarchical property which enables participants to move between various levels of negotiation, and the use of the nucleolus, a cooperative-game-solution concept first introduced by Schmeidler (1969), An evaluation of the strengths and weaknesses of game theory in this context forms the conclusion.

The generation algorithm for rectangular dissections given by Mitchell et al (1976) is shown not to be exhaustive, and a formal procedure is presented for the exhaustive generation of the class of ‘nonaligned’ rectangular dissections.