Some techniques using linear algebra was introduced by Faugore in F4 to speed up the reduction process during Grobner basis computations. These techniques can also be used in fast implementations of F5 and some other ...Some techniques using linear algebra was introduced by Faugore in F4 to speed up the reduction process during Grobner basis computations. These techniques can also be used in fast implementations of F5 and some other signature-based Grobner basis algorithms. When these techniques are applied, a very important step is constructing matrices from critical pairs and existing polynomials by the Symbolic Preprocessing function （given in F4）. Since multiplications of monomials and polynomials are involved in the Symbolic Preprocessing function, this step can be very costly when the number of involved polynomials/monomials is huge. In this paper, multiplications of monomials and polynomials for a Boolean polynomial ring are investigated and a specific method of implementing the Symbolic Preprocessing function over Boolean polynomial rings is reported. Many examples have been tested by using this method, and the experimental data shows that the new method is very efficient.展开更多
Rational Univariate Representation （RUR） of zero-dimensional ideals is used to describe the zeros of zero-dimensional ideals and RUR has been studied extensively. In 1999, Roullier proposed an efficient algorithm to...Rational Univariate Representation （RUR） of zero-dimensional ideals is used to describe the zeros of zero-dimensional ideals and RUR has been studied extensively. In 1999, Roullier proposed an efficient algorithm to compute RUR of zero-dimensional ideals. In this paper, we will present a new algorithm to compute Polynomial Univariate Representation （PUR） of zero-dimensional ideals. The new algorithm is based on some interesting properties of Grobner basis. The new algorithm also provides a method for testing separating elements.展开更多
A partition-of-unity （PU） based ＂FE-Meshfree＂ three-node triangular element （Trig3-RPIM） was recently developed for linear elastic problems. This Trig3-RPIM element employs hybrid shape functions that combine th...A partition-of-unity （PU） based ＂FE-Meshfree＂ three-node triangular element （Trig3-RPIM） was recently developed for linear elastic problems. This Trig3-RPIM element employs hybrid shape functions that combine the shape functions of three-node triangular element （Trig3） and radial-polynomial basis functions for the purpose of synergizing the merits of both finite element method and meshfree method. Although Trig3-RPIM element is capable of obtaining higher accuracy and convergence rate than the Trig3 element and four-node iso-parametric quadrilateral element without adding extra nodes or degrees of freedom （DOFs）, the nodal stress field through Trig3-RP1M element is not continuous and extra stress smooth operations are still needed in the post processing stage. To further improve the property of Trig3-RPIM element, a new PU-based triangular element with continuous nodal stress, called Trig3-RPIMcns, is developed. Numerical examples including several linear, free vibration and forced vibration test problems, have confirmed the correctness and feasibility of the proposed Trig3-RPIMcns element.展开更多
基金supported by the National Key Basic Research Program of China under Grant Nos.2013CB834203 and 2011CB302400the National Nature Science Foundation of China under Grant Nos.11301523,11371356,61121062+1 种基金the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No.XDA06010701IEE’s Research Project on Cryptography under Grant Nos.Y3Z0013102,Y3Z0018102,and Y4Z0061A02
文摘Some techniques using linear algebra was introduced by Faugore in F4 to speed up the reduction process during Grobner basis computations. These techniques can also be used in fast implementations of F5 and some other signature-based Grobner basis algorithms. When these techniques are applied, a very important step is constructing matrices from critical pairs and existing polynomials by the Symbolic Preprocessing function （given in F4）. Since multiplications of monomials and polynomials are involved in the Symbolic Preprocessing function, this step can be very costly when the number of involved polynomials/monomials is huge. In this paper, multiplications of monomials and polynomials for a Boolean polynomial ring are investigated and a specific method of implementing the Symbolic Preprocessing function over Boolean polynomial rings is reported. Many examples have been tested by using this method, and the experimental data shows that the new method is very efficient.
基金supported by National Key Basic Research Project of China(Grant No. 2011CB302400)National Natural Science Foundation of China (Grant Nos. 10971217,60821002/F02)
文摘Rational Univariate Representation （RUR） of zero-dimensional ideals is used to describe the zeros of zero-dimensional ideals and RUR has been studied extensively. In 1999, Roullier proposed an efficient algorithm to compute RUR of zero-dimensional ideals. In this paper, we will present a new algorithm to compute Polynomial Univariate Representation （PUR） of zero-dimensional ideals. The new algorithm is based on some interesting properties of Grobner basis. The new algorithm also provides a method for testing separating elements.
基金the National Natural Science Foundation of China(Grant Nos.51609240,11572009&51538001)and the National Basic Research Program of China(Grant No.2014CB047100)
文摘A partition-of-unity （PU） based ＂FE-Meshfree＂ three-node triangular element （Trig3-RPIM） was recently developed for linear elastic problems. This Trig3-RPIM element employs hybrid shape functions that combine the shape functions of three-node triangular element （Trig3） and radial-polynomial basis functions for the purpose of synergizing the merits of both finite element method and meshfree method. Although Trig3-RPIM element is capable of obtaining higher accuracy and convergence rate than the Trig3 element and four-node iso-parametric quadrilateral element without adding extra nodes or degrees of freedom （DOFs）, the nodal stress field through Trig3-RP1M element is not continuous and extra stress smooth operations are still needed in the post processing stage. To further improve the property of Trig3-RPIM element, a new PU-based triangular element with continuous nodal stress, called Trig3-RPIMcns, is developed. Numerical examples including several linear, free vibration and forced vibration test problems, have confirmed the correctness and feasibility of the proposed Trig3-RPIMcns element.
