We study ows in physical networks with a potential function deened over the nodes and a ow deened over the arcs. The networks have the further property that the ow on any arc a is a given increasing function of the diierence in potential between its initial and terminal node. An example is the equilibrium ow in water-supply pipe networks where the potential… (More)
A porous rectangular dam is above a horizontal impermeable base. There is a steady ow in which water seeps through the dam from one reservoir (on the left) to a lower reservoir (on the right). Because of gravity the water does not ow through the entire dam and the dam is dry near its upper right corner. The interface separating the dry and wet regions of… (More)
The torsion function of a plane domain is a function which is zero on the boundary of the domain and whose Laplacian is minus one at every point in the interior of the domain. We survey qualitative properties of the torsion function, and bounds on it and its derivatives, available when the plane domain is convex.
A graph is (m, k)-colourable if its vertices can be coloured with m colours such that the maximum degree of the subgraph induced on the set of all vertices receiving the same colour is at most k. The k-defective chromatic number χ k (G) is the least positive integer m for which graph G is (m, k)-colourable. Let f (m, k; planar) be the smallest order of a… (More)
In this paper Computer Aided Assessment (CAA) systems involving the delivery of questions across the web that are underpinned by Computer Algebra (CA) packages are discussed. This underpinning allows students to enter answers, have them parsed by the CA system, have them typechecked by the CA system, which are then passed through a marking procedure which… (More)
In a recent paper in this journal De La Grandville and Solow  presented a conjecture concerning Power Means. A counterexample to their conjecture is given.
Consider convex plane domains D(t) = (1 − t)D 0 + tD 1 , 0 ≤ t ≤ 1. We first prove that the 1/4-power of the polar moment of inertia about the centroid of D(t) is concave in t. From this we deduce some isoperimetric inequalities.
A graph is (m, k)-colourable if its vertices can be coloured with m colours such that the maximum degree of any subgraph induced on vertices receiving the same colour is at most k. The k-defective chromatic number for a graph is the least positive integer m for which the graph is (m, k)-colourable. All triangle-free graphs on 8 or fewer vertices are (2,… (More)