, 379-394), calculates lower bounds on the cost of using each node and edge in a feasible path using a single optimal Lagrange multiplier for the relaxation of the WCSPP. These lower bounds are used in conjunction with an upper bound to eliminate nodes and edges. However, for each node and edge, a Lagrangean dual problem exists whose solution may differ from the relaxation of the full problem. Thus, using one Lagrange multiplier does not offer the best possible network reduction. Furthermore, eliminating nodes and edges from the network may change the Lagrangean dual solutions in the remaining reduced network, warranting an iterative solution and reduction procedure. We develop a method for solving the related Lagrangean dual problems for each edge simultaneously which is iterated with eliminating nodes and edges. We demonstrate the effectiveness of our method computationally: we test it against several others and show that it both reduces solve time and the number of intractable problems encountered. We use a modified version of Carlyle and Wood's (38th Annual ORSNZ Conference, Hamilton, New Zealand, November, 2003) enumeration algorithm in the gap closing stage. We also make improvements to this algorithm and test them computationally.

This article describes the use of the SACI package-a package for calculating the energy levels and wavefunctions of a multi-electron quantum dot modelled as a 2D harmonic well with electrons interacting through a Coulomb potential and under the influence of a perpendicular magnetic field. ‡ Introduction Quantum dots are artificially fabricated atoms, in which charge carriers are confined in all three dimensions just like electrons in real atoms. Consequently, they exhibit properties normally associated with real atoms such as quantised energy levels and shell structures. These properties are described by the electron wavefunctions whose evolution is governed by the Schrödinger equation and the Pauli exclusion principle.There are many methods available to solve the Schrödinger equation for multiple electrons. They roughly fall into the categories of the diagonalisation method, mean-field density-functional theory, and the self-consistent field approach. One of the first theoretical studies of quantum dots was by Pfannkuche et al. [1], who compared the results of Hartree-Fock self-consistent calculations and exact diagonalisation of the Hamiltonian for two electrons in a circularly symmetric parabolic potential. They found good agreement between the two methods for the triplet state but marked differences for the singlet state, indicating important spin correlations were not included properly in their HartreeFock model. This suggests that the proper treatment of electron spins is crucial for correctly obtaining the electronic structures in quantum dots.Examples of self-consistent field approaches in the literature include Yannouleas and Landman [2,3], who studied circularly symmetric quantum dots using an unrestricted spin-space Hartree-Fock approach, and McCarthy et al. [4], who developed a Hartree-Fock Mathematica package. Macucci et al. [5] studied quantum dots with up to 24 electrons using a mean-field local-density-functionalThe Mathematica Journal 10:2

ABSTRACT:The exact diagonalization method using a spin-adapted basis is employed to calculate the electronic structure of a multi-electron quantum dot. By isolating spin and orbital angular momentum eigenstates, we have significantly reduced the size of the matrices required in comparison with the standard configuration interaction method. A novel approach to the simplification of the interaction integrals that arise in the calculation is also presented, which allows exact evaluation of the Hamiltonian matrix required in the calculations. This Mathematica package permits accurate calculation of energy levels and wave functions for both ground and excited states of multiple electrons confined in a circular quantum dot.

Abstract. In many path planning situations we would like to find a path of constrained Euclidean length in R 2 that minimises a line integral. We call this the Continuous Length-Constrained Minimum Cost Path Problem (C-LCMCPP). Generally, this will be a non-convex optimization problem, for which continuous approaches only ensure locally optimal solutions. However, network discretisations yield weight constrained network shortest path problems (WCSPPs), which can in practice be solved to global optimality, even for large networks; we can readily find a globally optimal solution to an approximation of the C-LCMCPP. Solutions to these WCSPPs yield feasible solutions and hence upper bounds. We show how networks can be constructed, and a WCSPP in these networks formulated, so that the solutions provide lower bounds on the global optimum of the continuous problem. We give a general convergence scheme for our network discretisations and use it to prove that both the upper and lower bounds so generated converge to the global optimum of the C-LCMCPP, as the network discretisation is refined. Our approach provides a computable lower bound formula (of course the upper bounds are readily computable). We give computational results showing the lower bound formula in practice, and compare the effectiveness of our network construction technique with that of standard grid-based approaches in generating good quality solutions. We find that for the same computational effort, we are able to find better quality solutions, particularly when the length-constraint is tighter.