The F5 Criterion Revised

The purpose of this work is to generalize part of the theory behind Faugere\u27s F5 algorithm. This is one of the fastest known algorithms to compute a Gröbner basis of a polynomial ideal I generated by polynomials f1,…,fm. A major reason for this is what Faugere called the algorithm\u27s new criterion, and we call the F5 criterion : it provides a sufficient condition for a set of polynomialsGto be a Gröbner basis. However. the F5 algorithm is difficult to grasp, and there are unresolved questions regarding its termination. This paper introduces some new concepts that place the criterion in a more general setting:S-Gröbner bases and primitive S-irreducible polynomials. We use these to propose a new, simple algorithm based on a revised F5 criterion. The new concepts also enable us to remove various restrictions, such as proving termination without the requirement that f1,…,fm be a regular sequence. (C) 2011 Elsevier Ltd. All rights reserved

The F5 Criterion Revised

The purpose of this work is to generalize part of the theory behind Faugere\u27s F5 algorithm. This is one of the fastest known algorithms to compute a Gröbner basis of a polynomial ideal I generated by polynomials f1,…,fm. A major reason for this is what Faugere called the algorithm\u27s new criterion, and we call the F5 criterion : it provides a sufficient condition for a set of polynomialsGto be a Gröbner basis. However. the F5 algorithm is difficult to grasp, and there are unresolved questions regarding its termination. This paper introduces some new concepts that place the criterion in a more general setting:S-Gröbner bases and primitive S-irreducible polynomials. We use these to propose a new, simple algorithm based on a revised F5 criterion. The new concepts also enable us to remove various restrictions, such as proving termination without the requirement that f1,…,fm be a regular sequence. (C) 2011 Elsevier Ltd. All rights reserved

Spectrophotometer-Integrating-Sphere System for Computing Solar Absorptance

A commercially available ultraviolet, visible, near-infrared spectrophotometer was modified to utilize an 8-inch-diameter modified Edwards-type integrated sphere. Software was written so that the reflectance spectra could be used to obtain solar absorptance values of 1-inch-diameter specimens. A descriptions of the system, spectral reflectance, and software for calculation of solar absorptance from reflectance data are presented

Diffusional Interactions at Dual Disk Microelectrodes: Comparison of Experiment with Three-dimensional Random Walk Simulations

Three dimensional random walk simulations were used to model the diffusional interactions between coplanar dual disk microelectrodes. Working curves for the dependence of shielding factor, collection efficiency, and amplification factor on the size of the gap between the electrodes were developed. The simulations showed that when the gap size was ≥22 radii, the electrodes could be considered as independent and diffusionally isolated. Empirical equations for determining the gap size from simple shielding and generator-collector experiments were established. The simulation was used to model the collector electrode response for square waves of various frequencies applied to the generator electrode. Simulation results were in excellent agreement with the experimental results for several dual disk electrodes having various gap sizes

XXXIV. The most economical potential-difference to employ with incandescent lamps

n/