Upper and lower bounds for the energy of bipartite graphs

Using Lagrange's multiplier rule, we find upper and lower bounds of the energy of a bipartite graph G, in terms of the number of vertices, edges and the spectral moment of fourth order. Moreover, the upper bound is attained in a graph G if and only if G is the graph of a symmetric balanced incomplete block design (BIBD). Also, we determine the graphs for which the lower bound is sharp.     

Møller-Plesset (MP2) perturbation theory for large molecules

Summary A novel formulation of MP2 theory is presented which starts from the Laplace transform MP2 ansatz, and subsequently moves from a molecular orbital (MO) representation to an atomic orbital (AO) representation. Consequently, the new formulation is denoted AO-MP2. As in traditional MP2 approaches electron repulsion integrals still need to be transformed. Strict bounds on the individual MP2 energy contribution of each intermediate four-index quantity allow to screen off numerically insignificant integrals with a single threshold parameter. Implicit in our formulation is a bound to two-particle density matrix elements. For small molecules the computational cost for AO-MP2 calculations is about a factor of 100 higher than for traditional MO-based approaches, but due to screening the computational effort in larger systems will only grow with the fourth power of the size of the system (or less) as is demonstrated both in theory and in application. MP2 calculations on (non-metallic) crystalline systems seem to be a feasible extension of the Laplace transform approach. In large molecules the AO-MP2 ansatz allows massively parallel MP2 calculations without input/output of four-index quantities provided that each processor has in-core memory for a limited number of two-index quantities. Energy gradient formulas for the AO-MP2 approach are derived.Dedicated to Prof. W. Kutzelnigg whose books on theoretical chemistry aroused my interest in this field     

Scheduling in network flow shops

We consider the general problem of static scheduling of a set of jobs in a network flow shop. In network flow shops, the scheduler not only has to sequence and schedule but also must concurrently determine the process routing of the jobs through the shop. In this paper, we establish the computational complexity of this new class of scheduling problem and propose a general purpose heuristic procedure. The performance of the heuristic is analyzed when makespan, cycle time and average flow time are the desired objectives.This research has been supported by the UCLA Academic Senate Grant #95.     

Computational aspects of two-segment separable programming

Recursive separable programming algorithms based on local, two-segment approximations are described for the solution of separable convex programs. Details are also given for the computation of lower bounds on the optimal value by both a primal and a dual approach, and these approaches are compared. Computational comparisons of the methods are provided for a variety of test problems, including a water supply application (with more than 600 constraints and more than 900 variables) and an econometric modelling problem (with more than 200 variables). Research supported by National Science Foundation Grants MCS74-20584 A02 and MCS-7901066.     

The Two-Dimensional Finite Bin Packing Problem. Part II: New lower and upper bounds

This paper is the second of a two part series and describes new lower and upper bounds for a more general version of the Two-Dimensional Finite Bin Packing Problem (2BP) than the one considered in Part I (see Boschetti and Mingozzi 2002). With each item is associated an input parameter specifying if it has a fixed orientation or it can be rotated by . This problem contains as special cases the oriented and non-oriented 2BP. The new lower bound is based on the one described in Part I for the oriented 2BP. The computational results on the test problems derived from the literature show the effectiveness of the new proposed lower and upper bounds.     

Chip firing and all-terminal network reliability bounds

The (all-terminal) reliability of a graph G is the probability that all vertices are in the same connected component, given that vertices are always operational but edges fail independently each with probability p. Computing reliability is #P-complete, and hence is expected to be intractable. Consequently techniques for efficiently (and effectively) bounding reliability have been the major thrust of research in the area. We utilize a deep connection between reliability and chip firings on graphs to improve previous bounds for reliability.     

On the Dirac equation in the algebraic approximation

It is shown that the restrictive conditions of Wood et al. [1] are not necessary to reach the conclusion that the Dirac hamiltonian, projected onto the space of the large component, exhibits variational properties. The eigenvalue spectrum of matrix approximations to the partitioned hamiltonian (obtained by matrix partitioning) converges to the exact spectrum in the limit of infinite order (assuming completeness) but not necessarily from above as for true matrix representations (obtained from operator partitioning). Optimization of non-linear parameters is shown not to cause variational instabilities.Control Data Corporation PACER fellow 1984–1986     

Structural results for partially observed control models

A general partially observed control model with discrete time parameter is investigated. Our main interest concerns monotonicity results and bounds for the value functions and for optimal policies. In particular, we show how the value functions depend on the observation kernels and we present conditions for a lower bound of an optimal policy. Our approach is based on two multivariate stochastic orderings: theTP ₂ ordering and the Blackwell ordering.Dedicated to Prof. Dr. K. Hinderer on the occassion of his 60^th birthday     

Microstructures réalisant toute compressibilité ou conductivité encadrée par les bornes de Hashin et Shtrikman

The conductivity or bulk modulus of an isotropic composite consisting of two well-ordered isotropic phases is situated between the lower and upper bounds of Hashin and Shtrikman. This work is concerned with the construction of two new microstructures capable of realizing every conductivity or bulk modulus comprised between these bounds. The two microstructures obtained, appearing to be simpler than those presented in the literature to achieve the same purpose, have the property that the variation of the corresponding conductivity or bulk modulus is strictly monotone with respect to the geometric parameter characterizing them. To cite this article: Q.-C. He, H.L. Quang, C. R. Mecanique 333 (2005).