This article describes the importance of detecting grain deviation caused by knots and reviews the main methods used in measuring grain orientation surrounding knots. It discusses the potential of using Diffusion Tens...This article describes the importance of detecting grain deviation caused by knots and reviews the main methods used in measuring grain orientation surrounding knots. It discusses the potential of using Diffusion Tensor Magnetic Resonance Imaging to track and map the grain deviation caused by knots.展开更多
It is one of the oldest research topics in computer algebra to determine the equivalence of Riemann tensor indexed polynomials. However, it remains to be a challenging problem since Grbner basis theory is not yet powe...It is one of the oldest research topics in computer algebra to determine the equivalence of Riemann tensor indexed polynomials. However, it remains to be a challenging problem since Grbner basis theory is not yet powerful enough to deal with ideals that cannot be finitely generated. This paper solves the problem by extending Grbner basis theory. First, the polynomials are described via an infinitely generated free commutative monoid ring. The authors then provide a decomposed form of the Grbner basis of the defining syzygy set in each restricted ring. The canonical form proves to be the normal form with respect to the Grbner basis in the fundamental restricted ring, which allows one to determine the equivalence of polynomials. Finally, in order to simplify the computation of canonical form, the authors find the minimal restricted ring.展开更多
基金The support of the New Zealand Foundation for Research,Science and Technology (Contract No. C04X0705) is gratefully acknowledged
文摘This article describes the importance of detecting grain deviation caused by knots and reviews the main methods used in measuring grain orientation surrounding knots. It discusses the potential of using Diffusion Tensor Magnetic Resonance Imaging to track and map the grain deviation caused by knots.
基金supported by the National Natural Science Foundation of China under Grant No.11701370the Natural Science Foundation of Shanghai under Grant No.15ZR1401600
文摘It is one of the oldest research topics in computer algebra to determine the equivalence of Riemann tensor indexed polynomials. However, it remains to be a challenging problem since Grbner basis theory is not yet powerful enough to deal with ideals that cannot be finitely generated. This paper solves the problem by extending Grbner basis theory. First, the polynomials are described via an infinitely generated free commutative monoid ring. The authors then provide a decomposed form of the Grbner basis of the defining syzygy set in each restricted ring. The canonical form proves to be the normal form with respect to the Grbner basis in the fundamental restricted ring, which allows one to determine the equivalence of polynomials. Finally, in order to simplify the computation of canonical form, the authors find the minimal restricted ring.
