In this paper, we prove the asymptotic growth rate of the summatory function of the pairwise maxima of the generalised divisor function d k (n), for a fixed positive integer k ≥ 2. This result generalises previous results of Kátai, Erdős and Hall on the local behaviour of divisor function on short intervals.

Abstract-Simulated annealing (SA) is a provably convergent optimiser for single-objective (SO) problems. Previously proposed MO extensions have mostly taken the form of an SO SA optimising a composite function of the objectives. We propose an MO SA utilising the relative dominance of a solution as the system energy for optimisation, eliminating problems associated with composite objective functions. We also propose a method for choosing perturbation scalings promoting search both towards and across the Pareto front.We illustrate the SA's performance on standard test problems. The new SA is shown to promote rapid convergence to the true Pareto front with a good coverage of points across it.

Simulated annealing is a provably convergent optimiser for single-objective problems. Previously proposed multiobjective extensions have mostly taken the form of a singleobjective simulated annealer optimising a composite function of the objectives. We propose a multi-objective simulated annealer utilising the relative dominance of a solution as the system energy for optimisation, eliminating problems associated with composite objective functions. We also propose a method for choosing perturbation scalings promoting search both towards and across the Pareto front.We illustrate the simulated annealer's performance on a suite of standard test problems and provide comparisons with another multi-objective simulated annealer and the NSGA-II genetic algorithm. The new simulated annealer is shown to promote rapid convergence to the true Pareto front with a good coverage of solutions across it comparing favourably with the other algorithms.An application of the simulated annealer to an industrial problem, the optimisation of a Code Division Multiple Access (CDMA) mobile telecommunications network's air interface, is presented and the simulated annealer is shown to generate non-dominated solutions with an even and dense coverage that outperform single objective genetic algorithm optimisers.

No abstract

Complete interpolating sequences for L 2 (−π, π) are considered under the condition that the real parts of the sequence are a subsequence of the scaled integers xZ, x > 0. It is found that this condition leads to very specific and restrictive conditions on the existence and structure of complete interpolating sequences for L 2 (−π, π). Further general results in the case of bunched sampling of Bernstein functions are also given.