A dual algorithm for the one-machine scheduling problem

A branch and bound algorithm is presented for the problem of schedulingn jobs on a single machine to minimize tardiness. The algorithm uses a dual problem to obtain a good feasible solution and an extremely sharp lower bound on the optimal objective value. To derive the dual problem we regard the single machine as imposing a constraint for each time period. A dual variable is associated with each of these constraints and used to form a Lagrangian problem in which the dualized constraints appear in the objective function. A lower bound is obtained by solving the Lagrangian problem with fixed multiplier values. The major theoretical result of the paper is an algorithm which solves the Lagrangian problem in a number of steps proportional to the product ofn ² and the average job processing time. The search for multiplier values which maximize the lower bound leads to the formulation and optimization of the dual problem. The bounds obtained are so sharp that very little enumeration or computer time is required to solve even large problems. Computational experience with 20-, 30-, and 50-job problems is presented.     

Very slowly varying functions

A real-valued functionf of a real variable is said to be?-slowly varying (?-s.v.) if lim _x→∞ ?(x) [f(x+α)?f(x)]=0 for each α. It is said to be uniformly?-slowly varying (u.?-s.v.) if lim _x→∞ sup _{α ∈ I} ?(x) |f(x+α)?f(x)|=0 for every bounded intervalI. It is supposed throughout that ? is positive and increasing. It is proved that if? increases rapidly enough, then every?-s.v. functionf must be u.?-s.v. and must tend to a limit at ∞. Regardless of the rate of increase of?, a measurable functionf must be u.?-s.v. if it is?-s.v. Examples of pairs (?,f) are given that illustrate the necessity for the requirements on? andf in these results.     

A theoretical investigation of charge transfer in several substituted acridinium ions

We present calculations for various properties of the ground and excited states of several arylamine-substituted acridinium ion systems that have been studied experimentally. Using ab initio and semiempirical quantum mechanical methods together with the generalized Mulliken-Hush (GMH) model, we examine the excitation energies, dipole moment shifts, and electronic coupling elements for the vertical charge shift (CSh) processes in these systems. We also examine solvent effects on these properties using a dielectric continuum reaction field model. The results are in generally good agreement with available experimental results and indicate that there is strong electronic coupling in these systems over a wide range of torsional angles. Nevetheless, the initial and final cationic states remain reasonably well-localized over this range, and thus TICT state formation is unlikely in these systems. Finally, a version of the GMH model based on Koopmans' Theorem is developed and found to yield coupling elements generally within a factor of 2 of the many-electron GMH for a sample acridinium system, but with overestimated adiabatic and diabatic dipole moment differences.     

Determining synthetic failures in

A combinatorial tripeptide library having the general form D-Glu-Xxx-Xxx-CONH2 has been synthesized using a standard mix and split synthetic protocol that is expected to produce 676 components. All components of the mixture were analyzed using a new high-resolution ion mobility/time-of-flight mass spectrometer coupled with an electrospray ionization source. In this approach ions are separated by differences in their gas-phase mobilities prior to being introduced into the mass spectrometer for mass-to-charge analysis. The peptide library includes a wide range of different sequence, structural, and stereo isomers; trends in the number of expected and resolved isomers that are observed at each m/z ratio allow specific synthetic steps that have failed to be identified, even in the presence of other isomers. Information about the relative abundances of different isomers should dramatically improve the reliability of binding affinity studies from direct analysis of mixtures.     

Comparing Positively and Negatively Charged Distonic Radical Ions in Phenylperoxyl Forming Reactions

In the gas phase, arylperoxyl forming reactions play a significant role in low-temperature combustion and atmospheric processing of volatile organic compounds. We have previously demonstrated the application of charge-tagged phenyl radicals to explore the outcomes of these reactions using ion trap mass spectrometry. Here, we present a side-by-side comparison of rates and product distributions from the reaction of positively and negatively charge tagged phenyl radicals with dioxygen. The negatively charged distonic radical ions are found to react with significantly greater efficiency than their positively charged analogues. The product distributions of the anion reactions favor products of phenylperoxyl radical decomposition (e.g., phenoxyl radicals and cyclopentadienone), while the comparable fixed-charge cations yield the stabilized phenylperoxyl radical. Electronic structure calculations rationalize these differences as arising from the influence of the charged moiety on the energetics of rate-determining transition states and reaction intermediates within the phenylperoxyl reaction manifold and predict that this influence could extend to intra-molecular charge-radical separations of up to 14.5 Å. Experimental observations of reactions of the novel 4-(1-carboxylatoadamantyl)phenyl radical anion confirm that the influence of the charge on both rate and product distribution can be modulated by increasing the rigidly imposed separation between charge and radical sites. These findings provide a generalizable framework for predicting the influence of charged groups on polarizable radicals in gas phase distonic radical ions.

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Graphical Abstract

     

A determination of sin2θ w and the lepton and quark electroweak couplings frome + e − data

Published data on lepton pair and quark pair final states ine ⁺ e ^? annihilation have been analysed in a self consistent way to yield values for the lepton and quark weak vector and axial vector couplings. Generation universality has been tested for the leptons and under the assumption of the standard model of the weak interaction, the parameter sin²θ _w has been determined separately for the lepton and quark sector. In the renormalisation scheme α,G _μ and sin²θ _w , the result for the lepton final states is sin²θ _w = 0.212±0.014 and for the hadron final states, sin²θ _w = 0.236±0.015. The combined result for this single parameter in the model is sin²θ _w = 0.223±0.011±0.007, corresponding tom _Z =93.0 _?1.8 ^+2.0 GeV.     

A compact automatic wavemeter for use with tunable infrared diode lasers

We have studied the magnetic-dipole and electric-quadrupole contributions to second-harmonic generation (SHG) of centro-symmetric media. It can be shown that the phenomenological parameters , , can be arranged to form an effective -tensor, which describes dipole-forbidden SHG as the effect of a surface layer upon excitation with a single plane electromagnetic wave. An experimental technique is proposed allowing a determination of the usually very small term due to magnetic-dipole interaction using coherent compensation of its contribution by a coverage with an appropriate dye monolayer.