Linear algebra has a very important application in physics and technical disciplines. This article conducted a questionnaire survey on the factors that affect the effect of linear algebra learning;the questionnaire co...Linear algebra has a very important application in physics and technical disciplines. This article conducted a questionnaire survey on the factors that affect the effect of linear algebra learning;the questionnaire contains several aspects of learning attitude, learning interest, learning methods, teaching methods, etc.;based on recycling data, cross chi-square test and multiple logistic regression analysis </span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">are </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">us</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">ed</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> to obtain the factors that affect the effect of linear algebra learning. The research results show that: learning methods, learning attitudes, teaching methods and elementary algebra basics are the main factors that affect the learning effect of linear algebra;among them, there are positive correlations between teaching methods, learning methods, learning attitudes and learning effects;teaching methods, learning methods 3. The three principal components of learning attitude are positively correlated. Based on the research and analysis, the following conclusions are drawn: finding a suitable learning method for the college students and maintaining a positive learning attitude are effective means to improve the linear algebra learning effect of the college students;in teaching, it is recommended to advance with the times, the teaching content and teaching methods innovate to stimulate students’ interest in learning</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">,</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> thus improv</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">ing</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> the learning effect of college students’ linear algebra courses.展开更多
To find out what knowledge in linear algebra is essential to non-mathematics students, a reverse tracking method was used. Based on practical problems likely to encountered by students in subsequent engineering course...To find out what knowledge in linear algebra is essential to non-mathematics students, a reverse tracking method was used. Based on practical problems likely to encountered by students in subsequent engineering courses, the minimum contents required has been determined. Rules are proposed to meet the background of most freshman students. An application oriented, easy to understand, computer based text book “Applied Popular Linear Algebra with MATLAB” [1] was published.展开更多
We propose a continuous analogy of Newton’s method with inner iteration for solving a system of linear algebraic equations. Implementation of inner iterations is carried out in two ways. The former is to fix the numb...We propose a continuous analogy of Newton’s method with inner iteration for solving a system of linear algebraic equations. Implementation of inner iterations is carried out in two ways. The former is to fix the number of inner iterations in advance. The latter is to use the inexact Newton method for solution of the linear system of equations that arises at each stage of outer iterations. We give some new choices of iteration parameter and of forcing term, that ensure the convergence of iterations. The performance and efficiency of the proposed iteration is illustrated by numerical examples that represent a wide range of typical systems.展开更多
Some techniques using linear algebra was introduced by Faugère in F4 to speed up the reduction process during Gr?bner basis computations.These techniques can also be used in fast implementations of F5 and some ot...Some techniques using linear algebra was introduced by Faugère in F4 to speed up the reduction process during Gr?bner basis computations.These techniques can also be used in fast implementations of F5 and some other signature-based Gr?bner 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
Let Mn be the algebra of all n × n complex matrices and gl(n, C) be the general linear Lie algebra, where n ≥ 2. An invertible linear map φ : gl(n, C) → gl(n, C) preserves solvability in both directions if bo...Let Mn be the algebra of all n × n complex matrices and gl(n, C) be the general linear Lie algebra, where n ≥ 2. An invertible linear map φ : gl(n, C) → gl(n, C) preserves solvability in both directions if both φ and φ-1 map every solvable Lie subalgebra of gl(n, C) to some solvable Lie subalgebra. In this paper we classify the invertible linear maps preserving solvability on gl(n, C) in both directions. As a sequence, such maps coincide with the invertible linear maps preserving commutativity on Mn in both directions.展开更多
Let P be a parabolic subalgebra of a general linear Lie algebra gl(n, F) over a field F, where n ≥ 3, F contains at least n different elements, and char(F) = 2. In this article, we prove that generalized derivations,...Let P be a parabolic subalgebra of a general linear Lie algebra gl(n, F) over a field F, where n ≥ 3, F contains at least n different elements, and char(F) = 2. In this article, we prove that generalized derivations, quasiderivations, and product zero derivations of P coincide, and any generalized derivation of P is a sum of an inner derivation, a central quasiderivation, and a scalar multiplication map of P. We also show that any commuting automorphism of P is a central automorphism, and any commuting derivation of P is a central derivation.展开更多
A surface model called the fibre bundle model and a 3D object model based on linear Lie algebra model are proposed. Then an algorithm of 3D object recognition using the linear Lie algebra models is presented. It is a ...A surface model called the fibre bundle model and a 3D object model based on linear Lie algebra model are proposed. Then an algorithm of 3D object recognition using the linear Lie algebra models is presented. It is a convenient recognition method for the objects which are symmetric about some axis. By using the presented algorithm, the representation matrices of the fibre or the base curve from only finite points of the linear Lie algebra model can be obtained. At last some recognition results of practicalities are given.展开更多
In this paper, some parallel algorithms are described for solving numerical linear algebra problems on Dawning-1000. They include matrix multiplication, LU factorization of a dense matrix, Cholesky factorization of a ...In this paper, some parallel algorithms are described for solving numerical linear algebra problems on Dawning-1000. They include matrix multiplication, LU factorization of a dense matrix, Cholesky factorization of a symmetric matrix, and eigendecomposition of symmetric matrix for real and complex data types. These programs are constructed based on fast BLAS library of Dawning-1000 under NX environment.Some comparison results under different parallel environments and implementing methods are also given for Cholesky factorization. The execution time, measured performance and speedup for each problem on Dawning-1000 are shown. For matrix multiplication and LU factorization, 1.86GFLOPS and 1.53GFLOPS are reached.展开更多
Let g be the general linear Lie algebra consisting of all n×n matrices over a field F and with the usual bracket operation[x,y]= xy - yx.An invertible map(?):g→g is said to preserve staircase subalgebras if it m...Let g be the general linear Lie algebra consisting of all n×n matrices over a field F and with the usual bracket operation[x,y]= xy - yx.An invertible map(?):g→g is said to preserve staircase subalgebras if it maps every staircase subalgebra to some staircase subalgebra of the same dimension.In this paper,we devote to giving an explicit description on the invertible maps on g that preserve staircase subalgebras.展开更多
It is shown that for a derivation f(x1o…oxj-1oxj+1o…xk)=k∑j=1x1o…oxu-1oxj+1o…oxkof(xj) on a JB*-algebra β, there exists a unique C-linear *-derivation D : β→β near the derivation.
Let (L, ≤, ∨, ∧) be a complete Heyting algebra. In this article, the linear system Ax = b over a complete Heyting algebra, where classical addition and multiplication operations are replaced by ∨ and ∧ respective...Let (L, ≤, ∨, ∧) be a complete Heyting algebra. In this article, the linear system Ax = b over a complete Heyting algebra, where classical addition and multiplication operations are replaced by ∨ and ∧ respectively, is studied. We obtain: (i) the necessary and suffcient conditions for S(A, b)≠■l; (ii) the necessary conditions for |S(A, b)| = 1. We also obtain the vector ∈ Ln and prove that it is the largest element of S(A, b) if S(A, b)≠■.展开更多
We show that the non-linear semi-quantum Hamiltonians which may be expressed as(whereis the set of generators of some Lie algebra and are the classical conjugated canonical variables) always close a partial semi Lie a...We show that the non-linear semi-quantum Hamiltonians which may be expressed as(whereis the set of generators of some Lie algebra and are the classical conjugated canonical variables) always close a partial semi Lie algebra under commutation and, because of this, it is always possible to integrate the mean values of the quantum degrees of freedom of the semi-quantum non-linear system in the fashion:(whereis the Maximum Entropy Principle density operator) and, so, these kind of Hamiltonians always have associated dynamic invariants which are expressed in terms of the quantum degrees of freedom’s mean values. Those invariants are useful to characterize the kind of dynamics (regular or irregular) the system displays given that they can be fixed by means of the initial conditions imposed on the semi-quantum non-linear system.展开更多
Based on the dynamic equation, the performance .functional and the system constraint equation of time-invariant discrete LQ control problem, the generalized Riccati equations of linear equality constraint system are o...Based on the dynamic equation, the performance .functional and the system constraint equation of time-invariant discrete LQ control problem, the generalized Riccati equations of linear equality constraint system are obtained according to the minimum principle, then a deep discussion about the above equations is given, and finally numerical example is shown in this paper.展开更多
In this work we derived and analyzed the stability structure of an order eight rational integrator wherein our numerator and denominator is 4 (i.e. m = n = 4) for the solution of problems in ordinary differential equa...In this work we derived and analyzed the stability structure of an order eight rational integrator wherein our numerator and denominator is 4 (i.e. m = n = 4) for the solution of problems in ordinary differential equations. The integrator was observed to be A-stable and also L-stable.展开更多
文摘Linear algebra has a very important application in physics and technical disciplines. This article conducted a questionnaire survey on the factors that affect the effect of linear algebra learning;the questionnaire contains several aspects of learning attitude, learning interest, learning methods, teaching methods, etc.;based on recycling data, cross chi-square test and multiple logistic regression analysis </span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">are </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">us</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">ed</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> to obtain the factors that affect the effect of linear algebra learning. The research results show that: learning methods, learning attitudes, teaching methods and elementary algebra basics are the main factors that affect the learning effect of linear algebra;among them, there are positive correlations between teaching methods, learning methods, learning attitudes and learning effects;teaching methods, learning methods 3. The three principal components of learning attitude are positively correlated. Based on the research and analysis, the following conclusions are drawn: finding a suitable learning method for the college students and maintaining a positive learning attitude are effective means to improve the linear algebra learning effect of the college students;in teaching, it is recommended to advance with the times, the teaching content and teaching methods innovate to stimulate students’ interest in learning</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">,</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> thus improv</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">ing</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> the learning effect of college students’ linear algebra courses.
文摘To find out what knowledge in linear algebra is essential to non-mathematics students, a reverse tracking method was used. Based on practical problems likely to encountered by students in subsequent engineering courses, the minimum contents required has been determined. Rules are proposed to meet the background of most freshman students. An application oriented, easy to understand, computer based text book “Applied Popular Linear Algebra with MATLAB” [1] was published.
文摘We propose a continuous analogy of Newton’s method with inner iteration for solving a system of linear algebraic equations. Implementation of inner iterations is carried out in two ways. The former is to fix the number of inner iterations in advance. The latter is to use the inexact Newton method for solution of the linear system of equations that arises at each stage of outer iterations. We give some new choices of iteration parameter and of forcing term, that ensure the convergence of iterations. The performance and efficiency of the proposed iteration is illustrated by numerical examples that represent a wide range of typical systems.
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 Faugère in F4 to speed up the reduction process during Gr?bner basis computations.These techniques can also be used in fast implementations of F5 and some other signature-based Gr?bner 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The NSF (2009J05005) of Fujian Provincea Key Project of Fujian Provincial Universities-Information Technology Research Based on Mathematics
文摘Let Mn be the algebra of all n × n complex matrices and gl(n, C) be the general linear Lie algebra, where n ≥ 2. An invertible linear map φ : gl(n, C) → gl(n, C) preserves solvability in both directions if both φ and φ-1 map every solvable Lie subalgebra of gl(n, C) to some solvable Lie subalgebra. In this paper we classify the invertible linear maps preserving solvability on gl(n, C) in both directions. As a sequence, such maps coincide with the invertible linear maps preserving commutativity on Mn in both directions.
基金supported by the National Natural Science Foundation of China(11101084,11071040)the Fujian Province Nature Science Foundation of China(2013J01005)
文摘Let P be a parabolic subalgebra of a general linear Lie algebra gl(n, F) over a field F, where n ≥ 3, F contains at least n different elements, and char(F) = 2. In this article, we prove that generalized derivations, quasiderivations, and product zero derivations of P coincide, and any generalized derivation of P is a sum of an inner derivation, a central quasiderivation, and a scalar multiplication map of P. We also show that any commuting automorphism of P is a central automorphism, and any commuting derivation of P is a central derivation.
基金Sponsored by the Ministry of Education Foundation of China(5220308)
文摘A surface model called the fibre bundle model and a 3D object model based on linear Lie algebra model are proposed. Then an algorithm of 3D object recognition using the linear Lie algebra models is presented. It is a convenient recognition method for the objects which are symmetric about some axis. By using the presented algorithm, the representation matrices of the fibre or the base curve from only finite points of the linear Lie algebra model can be obtained. At last some recognition results of practicalities are given.
文摘In this paper, some parallel algorithms are described for solving numerical linear algebra problems on Dawning-1000. They include matrix multiplication, LU factorization of a dense matrix, Cholesky factorization of a symmetric matrix, and eigendecomposition of symmetric matrix for real and complex data types. These programs are constructed based on fast BLAS library of Dawning-1000 under NX environment.Some comparison results under different parallel environments and implementing methods are also given for Cholesky factorization. The execution time, measured performance and speedup for each problem on Dawning-1000 are shown. For matrix multiplication and LU factorization, 1.86GFLOPS and 1.53GFLOPS are reached.
基金The NSF (11126121) of ChinaPh.D.Fund (B2010-93) of Henan Polytechnic University+1 种基金Natural Science Research Program (112300410120) of Science and Technology Department of Henan ProvinceNatural Science Research Program (2011B110016) of Education Department of Henan Province
文摘Let g be the general linear Lie algebra consisting of all n×n matrices over a field F and with the usual bracket operation[x,y]= xy - yx.An invertible map(?):g→g is said to preserve staircase subalgebras if it maps every staircase subalgebra to some staircase subalgebra of the same dimension.In this paper,we devote to giving an explicit description on the invertible maps on g that preserve staircase subalgebras.
基金Supported by Korea Research Foundation (KRF-2004-041-C00023).
文摘It is shown that for a derivation f(x1o…oxj-1oxj+1o…xk)=k∑j=1x1o…oxu-1oxj+1o…oxkof(xj) on a JB*-algebra β, there exists a unique C-linear *-derivation D : β→β near the derivation.
基金supported by the NNSF (10471035,10771056) of China
文摘Let (L, ≤, ∨, ∧) be a complete Heyting algebra. In this article, the linear system Ax = b over a complete Heyting algebra, where classical addition and multiplication operations are replaced by ∨ and ∧ respectively, is studied. We obtain: (i) the necessary and suffcient conditions for S(A, b)≠■l; (ii) the necessary conditions for |S(A, b)| = 1. We also obtain the vector ∈ Ln and prove that it is the largest element of S(A, b) if S(A, b)≠■.
文摘We show that the non-linear semi-quantum Hamiltonians which may be expressed as(whereis the set of generators of some Lie algebra and are the classical conjugated canonical variables) always close a partial semi Lie algebra under commutation and, because of this, it is always possible to integrate the mean values of the quantum degrees of freedom of the semi-quantum non-linear system in the fashion:(whereis the Maximum Entropy Principle density operator) and, so, these kind of Hamiltonians always have associated dynamic invariants which are expressed in terms of the quantum degrees of freedom’s mean values. Those invariants are useful to characterize the kind of dynamics (regular or irregular) the system displays given that they can be fixed by means of the initial conditions imposed on the semi-quantum non-linear system.
基金Supported by the National Basic Research Program of China （No. 2007CB311201）, the National Natural Science Foundation of China （No.60833008 No.60803149）, and the Foundation of Guangxi Key Laboratory of Information and Communication （No.20902）.
文摘Based on the dynamic equation, the performance .functional and the system constraint equation of time-invariant discrete LQ control problem, the generalized Riccati equations of linear equality constraint system are obtained according to the minimum principle, then a deep discussion about the above equations is given, and finally numerical example is shown in this paper.
文摘In this work we derived and analyzed the stability structure of an order eight rational integrator wherein our numerator and denominator is 4 (i.e. m = n = 4) for the solution of problems in ordinary differential equations. The integrator was observed to be A-stable and also L-stable.
