Simulated annealing and genetic algorithms for minimizing mean flow time in an open shop

This paper considers the problem of scheduling n jobs on m machines in an open shop environment so that the sum of completion times or mean flow time becomes minimal. It continues recent work by Bräsel et al. [H. Bräsel, A. Herms, M. Mörig, T. Tautenhahn, T. Tusch, F. Werner, Heuristic constructive algorithms for open shop scheduling to minmize mean flow time, European J. Oper. Res., in press (doi.10.1016/j.ejor.2007.02.057)] on constructive algorithms. For this strongly NP-hard problem, we present two iterative algorithms, namely a simulated annealing and a genetic algorithm. For the simulated annealing algorithm, several neighborhoods are suggested and tested together with the control parameters of the algorithm. For the genetic algorithm, new genetic operators are suggested based on the representation of a solution by the rank matrix describing the job and machine orders. Extensive computational results are presented for problems with up to 50 jobs and 50 machines, respectively. The algorithms are compared relative to each other, and the quality of the results is also estimated partially by a lower bound for the corresponding preemptive open shop problem. For most of the problems, the genetic algorithm is superior when fixing the same number of 30 000 generated solutions for each algorithm. However, in contrast to makespan minimization problems, where the focus is on problems with an equal number of jobs and machines, it turns out that problems with a larger number of jobs than machines are the hardest problems.     

A littlewood-paley inequality for the Carleson operator

The Carleson operator is closely related to the maximal partial sum operator for Fourier series. We study generalizations of this operator in one and several variables.     

Rational points of bounded height on projective surfaces

We give upper bounds for the number of rational points of bounded height on the complement of the lines on projective surfaces.     

Convergence of the Magnus Series

The Magnus series is an infinite series which arises in the study of linear ordinary differential equations. If the series converges, then the matrix exponential of the sum equals the fundamental solution of the differential equation. The question considered in this paper is: When does the series converge? The main result establishes a sufficient condition for convergence, which improves on several earlier results.     

Path integral approach to quantum diffusion in metals

A general formalism for a quantum particle moving in an environment based on the path integral formulation of quantum mechanics is presented. The environment can be the valence electron gas of a metal or the phonons of the host in which the particle moves, or both. The advantage of this approach is that it gives a direct space-time picture of the motion. Integrating out the coordinates of the environment an effective action for the particle is obtained. It consist of two parts. An adiabatic term, that is local in time (and as such describable as a potential), describing the influence of the fully relaxed environment on the particle, and a term that accounts for the dynamical part of the response of the environment due to the motion of the particle. As an example I have considered a particle moving in a tight binding band with an electron gas environment. Three energy scales play a role here. They are the bare hopping amplitude, the temperature and the inverse tunneling time (i.e. the time it takes the particle to tunnel from one site to the next). Note that the Fermi energy drops out of the problem. I find, as did Kondo in his recent theory, a decreasing diffusion constant as a function of temperature.     

Sparse grids and hybrid methods for the chemical master equation

The direct numerical solution of the chemical master equation (CME) is usually impossible due to the high dimension of the computational domain. The standard method for solution of the equation is to generate realizations of the chemical system by the stochastic simulation algorithm (SSA) by Gillespie and then taking averages over the trajectories. Two alternatives are described here using sparse grids and a hybrid method. Sparse grids, implemented as a combination of aggregated grids are used to address the curse of dimensionality of the CME. The aggregated components are selected using an adaptive procedure. In the hybrid method, some of the chemical species are represented macroscopically while the remaining species are simulated with SSA. The convergence of variants of the method is investigated for a growing number of trajectories. Two signaling cascades in molecular biology are simulated with the methods and compared to SSA results. AMS subject classification (2000)  65C20, 60J25, 92C45