This paper takes the basic mathematical operations required to manipulate the cognitive maps. This paper start by presenting all the values that a causal relationship can take. By the using of causal algebra, cognitiv...This paper takes the basic mathematical operations required to manipulate the cognitive maps. This paper start by presenting all the values that a causal relationship can take. By the using of causal algebra, cognitive map (CC) become not only a graphical representation of a person’s beliefs, an agent or a particular area but also can capture the causal relationships existing between the concepts of a given system in a simple manner. Cognitive maps do not use a conventional algebra;algebra causal is needed to treat.展开更多
Let G be a generalized matrix algebra over a commutative ring R and Z（G） be the center of G. Suppose that F, T ：G→G are two co-commuting R-linear mappings, i.e., F（x）x = xT（x） for all x ∈ G. In this note, we ...Let G be a generalized matrix algebra over a commutative ring R and Z（G） be the center of G. Suppose that F, T ：G→G are two co-commuting R-linear mappings, i.e., F（x）x = xT（x） for all x ∈ G. In this note, we study the question of when co-commuting mappings on G are proper.展开更多
A map φ on a Lie algebra g is called to be commuting if [φ(x), x] = 0 for all x ∈ g. Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0, P a parabolic subalgeb...A map φ on a Lie algebra g is called to be commuting if [φ(x), x] = 0 for all x ∈ g. Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0, P a parabolic subalgebra of L. In this paper, we prove that a linear mapφon P is commuting if and only if φ is a scalar multiplication map on P.展开更多
Let g be the general linear Lie algebra consisting of all n x n matrices over a field F and with the usual bracket operation {x, y} =xy - yx. An invertible map φ ： g →g is said to preserve staircase subalgebras if ...Let g be the general linear Lie algebra consisting of all n x n matrices over a field F and with the usual bracket operation {x, y} =xy - yx. An invertible map φ ： 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
A property(C) for permutation pairs is introduced. It is shown that if a pair{π_1, π_2} of permutations of(1,2,…,n) has property(C),then the D-type map Φ_(π_1,π_2) on n× n complex matrices constructed from ...A property(C) for permutation pairs is introduced. It is shown that if a pair{π_1, π_2} of permutations of(1,2,…,n) has property(C),then the D-type map Φ_(π_1,π_2) on n× n complex matrices constructed from {π_1,π_2} is positive. A necessary and sufficient condition is obtained for a pair {π_1,π_2} to have property(C),and an easily checked necessary and sufficient condition for the pairs of the form {π~p,π~q} to have property(C) is given, whereπ is the permutation defined by π(i) = i + 1 mod n and 1≤ p < q≤ n.展开更多
For a Banach algebra A, we denote by .A＊ and .A＊＊ the first and the second duals of A respectively. Let T be a mapping from .A＊ to itself. In this article, we will investigate some stability results concerning the...For a Banach algebra A, we denote by .A＊ and .A＊＊ the first and the second duals of A respectively. Let T be a mapping from .A＊ to itself. In this article, we will investigate some stability results concerning the equations T（αf ＋ βg） -= αT（f） ＋ βT（g）, T（af） = aT（f） andT（αf ＋βg） ＋ T（αf - βg） =- 2α2T（f） ＋ 2β2T（g） where f, g e .A＊, a ∈ A, and α,β ∈ Q ／ {0}.展开更多
Let Q be the quaternion division algebra over real field F.Denote by H_n(Q)the set of all n×n hermitian matrices over Q.We characterize the additive maps from H_n(Q) into H_m(Q)that preserve rank-1 matrices when ...Let Q be the quaternion division algebra over real field F.Denote by H_n(Q)the set of all n×n hermitian matrices over Q.We characterize the additive maps from H_n(Q) into H_m(Q)that preserve rank-1 matrices when the rank of the image of I_n is equal to n. Let Q_R be the quaternion division algebra over the field of real number R.The additive maps from H_n(Q_R) into H_m(Q_R)that preserve rank-1 matrices are also given.展开更多
The Bijective Pattern Map maps the 3D surface patch to the 2D flat pattern. The bijection property has been proved. However, since the 3D Algebraic Mannequin is not developeable, meaning that it is not isometric to th...The Bijective Pattern Map maps the 3D surface patch to the 2D flat pattern. The bijection property has been proved. However, since the 3D Algebraic Mannequin is not developeable, meaning that it is not isometric to the 2D plane, the intrinsic geometry of the 3D surface patches is different from their images on the 2D plane. Consequently, distortion exists. The distortional property of the Bijective Pattern Map is discussed in this article.展开更多
This is the part n of the study of distortion property of the Bijective Pattern Map maps. In part I, the generic distortion was presented. In this article, the distortion using different set of reference points and pr...This is the part n of the study of distortion property of the Bijective Pattern Map maps. In part I, the generic distortion was presented. In this article, the distortion using different set of reference points and pre - image is compared with the result in part I. Moreover, the distortion induced by a change of reference is illustrated with examples. The formulation of the relationship between the distortion and the Gaussian curvature will be followed.展开更多
文摘This paper takes the basic mathematical operations required to manipulate the cognitive maps. This paper start by presenting all the values that a causal relationship can take. By the using of causal algebra, cognitive map (CC) become not only a graphical representation of a person’s beliefs, an agent or a particular area but also can capture the causal relationships existing between the concepts of a given system in a simple manner. Cognitive maps do not use a conventional algebra;algebra causal is needed to treat.
文摘Let G be a generalized matrix algebra over a commutative ring R and Z（G） be the center of G. Suppose that F, T ：G→G are two co-commuting R-linear mappings, i.e., F（x）x = xT（x） for all x ∈ G. In this note, we study the question of when co-commuting mappings on G are proper.
基金Supported by the National Natural Science Foundation of China（Ill01084） Supported by the Fujian Province Natural Science Foundation of China
文摘A map φ on a Lie algebra g is called to be commuting if [φ(x), x] = 0 for all x ∈ g. Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0, P a parabolic subalgebra of L. In this paper, we prove that a linear mapφon P is commuting if and only if φ is a scalar multiplication map on P.
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 x n matrices over a field F and with the usual bracket operation {x, y} =xy - yx. An invertible map φ ： 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partially supported by National Natural Science Foundation of China(11671294)
文摘A property(C) for permutation pairs is introduced. It is shown that if a pair{π_1, π_2} of permutations of(1,2,…,n) has property(C),then the D-type map Φ_(π_1,π_2) on n× n complex matrices constructed from {π_1,π_2} is positive. A necessary and sufficient condition is obtained for a pair {π_1,π_2} to have property(C),and an easily checked necessary and sufficient condition for the pairs of the form {π~p,π~q} to have property(C) is given, whereπ is the permutation defined by π(i) = i + 1 mod n and 1≤ p < q≤ n.
文摘For a Banach algebra A, we denote by .A＊ and .A＊＊ the first and the second duals of A respectively. Let T be a mapping from .A＊ to itself. In this article, we will investigate some stability results concerning the equations T（αf ＋ βg） -= αT（f） ＋ βT（g）, T（af） = aT（f） andT（αf ＋βg） ＋ T（αf - βg） =- 2α2T（f） ＋ 2β2T（g） where f, g e .A＊, a ∈ A, and α,β ∈ Q ／ {0}.
文摘Let Q be the quaternion division algebra over real field F.Denote by H_n(Q)the set of all n×n hermitian matrices over Q.We characterize the additive maps from H_n(Q) into H_m(Q)that preserve rank-1 matrices when the rank of the image of I_n is equal to n. Let Q_R be the quaternion division algebra over the field of real number R.The additive maps from H_n(Q_R) into H_m(Q_R)that preserve rank-1 matrices are also given.
文摘The Bijective Pattern Map maps the 3D surface patch to the 2D flat pattern. The bijection property has been proved. However, since the 3D Algebraic Mannequin is not developeable, meaning that it is not isometric to the 2D plane, the intrinsic geometry of the 3D surface patches is different from their images on the 2D plane. Consequently, distortion exists. The distortional property of the Bijective Pattern Map is discussed in this article.
文摘This is the part n of the study of distortion property of the Bijective Pattern Map maps. In part I, the generic distortion was presented. In this article, the distortion using different set of reference points and pre - image is compared with the result in part I. Moreover, the distortion induced by a change of reference is illustrated with examples. The formulation of the relationship between the distortion and the Gaussian curvature will be followed.
