Lower Bounds from State Space Relaxations for Concave Cost Network Flow Problems

In this paper we obtain Lower Bounds (LBs) to concave cost network flow problems. The LBs are derived from state space relaxations of a dynamic programming formulation, which involve the use of non-injective mapping functions guaranteing a reduction on the cardinality of the state space. The general state space relaxation procedure is extended to address problems involving transitions that go across several stages, as is the case of network flow problems. Applications for these LBs include: estimation of the quality of heuristic solutions; local search methods that use information of the LB solution structure to find initial solutions to restart the search (Fontes et al., 2003, Networks, 41, 221–228); and branch-and-bound (BB) methods having as a bounding procedure a modified version of the LB algorithm developed here, (see Fontes et al., 2005a). These LBs are iteratively improved by penalizing, in a Lagrangian fashion, customers not exactly satisfied or by performing state space modifications. Both the penalties and the state space are updated by using the subgradient method. Additional constraints are developed to improve further the LBs by reducing the searchable space. The computational results provided show that very good bounds can be obtained for concave cost network flow problems, particularly for fixed-charge problems.     

Embeddings and hyperplanes of the Lie incidence geometry of type E7,1

This paper continues a program to show that for most of the standard Lie incidence geometries, all geometric hyperplanes arise from a necessarily absolutely universal embedding, by addingE _7,1 to the list. It follows from [5, 12] that any projective embedding of this point line geometry is a homomorphic image of the one afforded by the 56-dimensional module for the groupE ₇(K).This work was supported by a grant from the National Science Foundation.     

Ernest Julius Wilczynski (in Memoriam)

Ringrazio il collega dell'Università di Chicago di avermi permesso di pubblicare qui un breve sunto della commemorazione da lui scritta; e sono dolente che lo spazio concessomi non me ne abbia permesso la pubblicazione integrale.Guido Fubini     

Excimer Laser Processing of Grey Cast Iron: Microstructural and Tribological Characterization

Pearlitic grey iron was treated by means of a XeCl laser to remelt only a shallow layer. This process allows to modify the heteregeneous surfaces of cast irons in order to improve their tribological behaviour. The energy density ranged from 0.3 to 3 J/cm² and the number of shots from 4 to 40. It is shown that after laser irradiation austenite is formed in the melted layer; compressive stresses at the surface were exhibited as well as the deposition of graphite and the formation of carbides. It is shown that excimer laser processing highly decreases the coefficient of friction and the wear rate. This revised version was published online in August 2006 with corrections to the Cover Date.     

2-Oxocephems. Synthesis and Properties of 7-Acylamino-2-oxocephem-4-carboxylic Acids

A synthesis of (6 R, 7 R)-7-acylamino-2-oxocephem-4-carboxylic acids from penicillins is reported based on the formation of the 2-oxocephem system in an intramolecular Wittig condensation. An interesting steric effect of the acylamino grouping on the course of this key step was used to advantage. Some physical and biological properties of this new type of Δ³-cephem are described.     

Chemical potential for harmonically interacting particles in a harmonic potential

The chemical potential μ, as it appears in density functional theory, is examined extensively for harmonically interacting spin-½ fermions in three dimensions. For this system the energy and chemical potential are discontinuous functions of the particle number if the most straightforward equation is used to define the energy for a noninteger particle number.     

Rotation and inversion barriers in 2-lithio-2-phenyl-1,3-dithianes

The barriers to phenyl rotation in 2-lithio-2-phenyl-cis-4,6-dimethyl-, 2-lithio-2-phenyl-4,4,6-trimethyl- and 2-lithio-2-phenyl-trans-4,6-dimethyl-1,3-dithiane are compared in tetrahydrofuran (THF) and hexamethylphosphortriamide (HMPA). In the first two cases, the barriers in THF are lower than those in HMPA, presumably because the lithio compound exists as a tight ion pair in THF but as a solvent-separated ion pair (with more delocalization of charge into the phenyl ring) in HMPA. However, in the trans-4,6-dimethyl compound the barriers are the same in the two solvents and nearly equal to the barriers for ring reversal. It is concluded that in this compound the rate-determining step for phenyl rotation may actually be ring reversal, at least in solvent HMPA.