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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.
Consider positive solutions of the one-dimensional heat equation. The space variable x lies in (?a; a): the time variable t in (0; 1). When the solution u satisses (i) u(a; t) = 0, and (ii) u(:; 0) is logconcave, we give a new proof based on the Maximum Principle, that, for any xed t > 0, u(:; t) remains logconcave. The same proof techniques are used to… (More)
SUMMARY Flow resistance laws, as used for example in water-supply pipe networks, are formulae relating the volume ow rate q along a pipe to the pressure-head diierence t between its ends, q = (t). is monotonic. The simple Hazen-Williams power \law" is often used: it has been claimed that the more complicated Colebrook-White law (CW) better represents… (More)