A better step-off algorithm for the knapsack problem

The knapsack problem, maximize Σ^m_{i = 1}c_ix_i when Σ^m_{i = 1}a_ix_i?b for integers x_i?0, can be solved by the classical step-off algorithm. The algorithm develops a series of feasible solutions with ever-increasing objective values. We make a change in the problem so that the step-off algorithm produces a series of solutions of not necessarily increasing objective values. A point is reached when no better solutions can be found and the calculation is stopped.     

Analysis of a statistical multiplexer with generalized periodic sources

We provide solution techniques for the analysis of multiplexers with periodic arrival streams, which accurately account for the effects of active and idle periods and of gradual arrival. In the models considered in this paper, it is assumed that each source alternates (periodically) between active and idle periods of fixed durations. Incoming packets are transmitted on the network link and excess information is stored in the multiplexing buffer when the aggregate input rate exceeds the capacity of the link. We are interested in the probability distribution of the buffer content for a given network link speed as a function of the number of sources and their characteristics, i.e., rate and duration of idle and active periods. We derive this distribution from two models: discrete time and continuous time systems. Discrete time systems operate in a slotted fashion, with a slot defining the base unit for data generation and transmission. In particular, in each slot the link is capable of transmitting one data unit and conversely an active source generates one data unit in that time. The continuous time model of the paper falls in the category of fluid models. Compared to previous works we allow a more general model for the periodic packet arrival process of each source. In discrete time, this means that the active period of a source can now extend over several consecutive slots instead of a single slot as in previous models. In continuous time, packet arrivals are not required to be instantaneous, but rather the data generation process can now take place over the entire duration of the active period. In both cases, these generalizations allow us to account for the progressive arrival of source data as a function of both the source speed and the amount of data it generates in an active period.This work was done while at the IBM T.J. Watson Research Center.This work was done while at the IBM T.J. Watson Research Center.Part of the work was done while visiting the IBM T.J. Watson Research Center.     

Spectral Analysis for Systems of Atoms and Molecules Coupled to the Quantized Radiation Field

We consider systems of static nuclei and electrons – atoms and molecules – coupled to the quantized radiation field. The interactions between electrons and the soft modes of the quantized electromagnetic field are described by minimal coupling, p→p−e A (x), where A(x) is the electromagnetic vector potential with an ultraviolet cutoff. If the interactions between the electrons and the quantized radiation field are turned off, the atom or molecule is assumed to have at least one bound state. We prove that, for sufficiently small values of the fine structure constant α, the interacting system has a ground state corresponding to the bottom of its energy spectrum. For an atom, we prove that its excited states above the ground state turn into metastable states whose life-times we estimate. Furthermore the energy spectrum is absolutely continuous, except, perhaps, in a small interval above the ground state energy and around the threshold energies of the atom or molecule. Received: 3 September 1998 / Accepted: 17 March 1999     

Bent Phosphaallenes With “Hidden” Lone Pairs as Ligands

Phosphaheteroallenes R−P=C=L, with L = N-heterocyclic carbenes (NHCs), can be viewed to a certain extent as phosphaisonitriles stabilized with NHCs, R−P=C:←L. The suitability of these molecules as ligands for coinage-metal ions was investigated and coordination through the central carbon center was observed in most cases. A combination of experiments, spectroscopic methods, and DFT calculations indicates the presence of a hidden electron pair at the carbon center of R−P=C:←L. Remarkably, this lone pair also inserts intramolecularly in C−H bonds showing the carbene-type reactivity which is expected for phosphaisonitriles.     

Impact of C=C/B−N Replacement on the Diels–Alder Reactivity of Curved Polycyclic Aromatic Hydrocarbons

The influence of the replacement of C=C bonds by isoelectronic B−N moieties on the reactivity of π-curved polycyclic aromatic hydrocarbons has been computationally explored by means of density functional theory calculations. To this end, we selected the Diels–Alder cycloaddition reactions of the parent corannulene and its BN-doped counterparts with either cyclopentadiene or maleic anhydride. In addition, the analogous reactions involving larger buckybowls, such as BN-hemifullerene, BN-circumtrindene, and BN-fullerene, have been also considered. It has been found that whereas corannulene behaves as a dienophile, its BN counterpart better acts as a diene. In contrast, the larger BN-curved systems cannot be used as dienes in Diels–Alder reactions, but undergo facile (i.e., low barrier) cycloaddition reactions with cyclopentadiene. The observed trends in reactivity, which cannot be directly explained by using typical frontier molecular orbital arguments, are quantitatively described in detail by means of state-of-the-art computational methods, namely the activation strain model of reactivity combined with the energy decomposition analysis method. The results of our calculations highlight the crucial role of the curvature of the system on the reactivity and its influence on the strength of the orbital interactions between the deformed reactants during their transformations.     

A Germylene Supported by Two 2-Pyrrolylphosphane Groups as Precursor to PGeP Pincer Square-Planar Group 10 Metal(II) and T-Shaped Gold(I) Complexes

An efficient synthesis of 2-di-tert-butylphosphanylmethylpyrrole (HpyrmPtBu₂), by treating 2-dimethylaminomethylpyrrole (HpyrmNMe₂) with tBu₂PH at 135 °C in the absence of any solvent, has allowed the preparation of the new PGeP germylene Ge(pyrmPtBu₂)₂ ( 1 ), by treating [GeCl₂(dioxane)] with LipyrmPtBu₂, in which the Ge atom is stabilized by intramolecular interactions with one (solid state) or both (solution) of its phosphane groups. Reactions of germylene 1 with Group 10 metal dichlorido complexes containing easily displaceable ligands have led to [MCl{κ³P,Ge,P-GeCl(pyrmPtBu₂)₂}] [M=Ni ( 2 ), Pd ( 3 ), Pt ( 4 )], which have an unflawed square-planar metal environment. Treatment of germylene 1 with [AuCl(tht)] (tht=tetrahydrothiophene) rendered [Au{κ³P,Ge,P-GeCl(pyrmPtBu₂)₂}] ( 5 ), which is a rare case of a T-shaped gold(I) complex. The hydrolysis of 5 gave the linear gold(I) derivative [Au(κP-HpyrmPtBu₂)₂]Cl ( 6 ). Complexes 2 – 5 contain a PGeP pincer chloridogermyl ligand that arises from the insertion of the Ge atom of germylene 1 into a M−Cl bond of the corresponding metal reagent. The bonding in these molecules has been studied by DFT/NBO/QTAIM calculations. These results demonstrate that the great flexibility of germylene 1 makes it a better precursor to PGeP pincer complexes than the previously known germylenes of this type.