An Improvement over the GVW Algorithm for Inhomogeneous Polynomial Systems

The GVW algorithm is a signature-based algorithm for computing Gr\"obner bases. If the input system is not homogeneous, some J-pairs with higher signatures but lower degrees are rejected by GVW's Syzygy Criterion, instead, GVW have to compute some J-pairs with lower signatures but higher degrees. Consequently, degrees of polynomials appearing during the computations may unnecessarily grow up higher and the computation become more expensive. In this paper, a variant of the GVW algorithm, called M-GVW, is proposed and mutant pairs are introduced to overcome inconveniences brought by inhomogeneous input polynomials. Some techniques from linear algebra are used to improve the efficiency. Both GVW and M-GVW have been implemented in C++ and tested by many examples from boolean polynomial rings. The timings show M-GVW usually performs much better than the original GVW algorithm when mutant pairs are found. Besides, M-GVW is also compared with intrinsic Gr\"obner bases functions on Maple, Singular and Magma. Due to the efficient routines from the M4RI library, the experimental results show that M-GVW is very efficient

Conductance through a single impurity in the metallic zigzag carbon nanotube

We investigate transport through a single impurity in metallic zigzag carbon nanotube and find the conductance sensitively depends on the impurity strength and the bias voltage. It is rather interesting that interplay between the current-carrying scattering state and evanescent modes leads to rich phenomena including resonant backward scattering, perfect tunneling and charge accumulations. In addition, we also find a dual relation between the backscattered conductance and the charge accumulation. At the end, relevance to the experiments is discussed.Comment: 3 pages, 3 figure