This paper presents the dual specification of the least-squares method. In other words, while the traditional (primal) formulation of the method minimizes the sum of squared residuals (noise), the dual specification m...This paper presents the dual specification of the least-squares method. In other words, while the traditional (primal) formulation of the method minimizes the sum of squared residuals (noise), the dual specification maximizes a quadratic function that can be interpreted as the value of sample information. The two specifications are equivalent. Before developing the methodology that describes the dual of the least-squares method, the paper gives a historical perspective of its origin that sheds light on the thinking of Gauss, its inventor. The least-squares method is firmly established as a scientific approach by Gauss, Legendre and Laplace within the space of a decade, at the beginning of the nineteenth century. Legendre was the first author to name the approach, in 1805, as “méthode des moindres carrés”, a “least-squares method”. Gauss, however, used the method as early as 1795, when he was 18 years old. Again, he adopted it in 1801 to calculate the orbit of the newly discovered planet Ceres. Gauss published his way of looking at the least-squares approach in 1809 and gave several hints that the least-squares algorithm was a minimum variance linear estimator and that it was derivable from maximum likelihood considerations. Laplace wrote a very substantial chapter about the method in his fundamental treatise on probability theory published in 1812.展开更多
The purpose of this paper is to introduce second order (K, F)-pseudoconvex and second order strongly (K, F)- pseudoconvex functions which are a generalization of cone-pseudoconvex and strongly cone-pseudoconvex functi...The purpose of this paper is to introduce second order (K, F)-pseudoconvex and second order strongly (K, F)- pseudoconvex functions which are a generalization of cone-pseudoconvex and strongly cone-pseudoconvex functions. A pair of second order symmetric dual multiobjective nonlinear programs is formulated by using the considered functions. Furthermore, the weak, strong and converse duality theorems for this pair are established. Finally, a self duality theorem is given.展开更多
As known, the method to obtain a sequence space by using convergence field of an infinite matrix is an old method in the theory of sequence spaces. However, the study of convergence field of an infinite matrix in the ...As known, the method to obtain a sequence space by using convergence field of an infinite matrix is an old method in the theory of sequence spaces. However, the study of convergence field of an infinite matrix in the space of almost convergent sequences is so new （see [15]）. The purpose of this paper is to introduce the new spaces ^ ～f and fo consisting of all sequences whose Ceshro transforms of order one are in the spaces f and ^ ～ f0, respectively. Also, in this paper, we show that ^ ～f and ^ ～f0 are linearly isomorphic to the spaces f and f0, respectively. The β- and γ-duals of the spaces ^ ～f and 2% are computed. Furthermore, the classes （^ ～f： μ） and （μ ： f） of infinite matrices are characterized for any given sequence space μ, and determined the necessary and sufficient conditions on a matrix A to satisfy Bc-core（Ax） K-core（x）, K-core（Ax） Bg-core（x）, Bc-core（Ax） Be-core（x）, Bc-core（Ax） t-core（x） for all x ∈ t∞.展开更多
In this paper, the extremal problem, min, of two convex bodies K and L in ?n is considered. For K to be in extremal position in terms of a decomposition of the identity, give necessary conditions together with the opt...In this paper, the extremal problem, min, of two convex bodies K and L in ?n is considered. For K to be in extremal position in terms of a decomposition of the identity, give necessary conditions together with the optimization theorem of John. Besides, we also consider the weaker optimization problem: min. As an application, we give the geometric distance between the unit ball B2n and a centrally symmetric convex body K.展开更多
Several general open mapping theorems for continuous linear maps, weakly continuous linear maps and closed linear maps are established respectively. From these general forms, a lot of specific open mapping theorems ca...Several general open mapping theorems for continuous linear maps, weakly continuous linear maps and closed linear maps are established respectively. From these general forms, a lot of specific open mapping theorems can be deduced.展开更多
In this paper, the author establishs general open mapping theorems for continuous, weakly continuous, closed and weakly singular linear maps respectively. From these general theorems, he deduces a lot of specific open...In this paper, the author establishs general open mapping theorems for continuous, weakly continuous, closed and weakly singular linear maps respectively. From these general theorems, he deduces a lot of specific open mapping theorems, which include some well-known theorems and some new interesting results. Particularly the author gives the extensions of Husain's open mapping theorem and Adasch's open mapping theorem.展开更多
文摘This paper presents the dual specification of the least-squares method. In other words, while the traditional (primal) formulation of the method minimizes the sum of squared residuals (noise), the dual specification maximizes a quadratic function that can be interpreted as the value of sample information. The two specifications are equivalent. Before developing the methodology that describes the dual of the least-squares method, the paper gives a historical perspective of its origin that sheds light on the thinking of Gauss, its inventor. The least-squares method is firmly established as a scientific approach by Gauss, Legendre and Laplace within the space of a decade, at the beginning of the nineteenth century. Legendre was the first author to name the approach, in 1805, as “méthode des moindres carrés”, a “least-squares method”. Gauss, however, used the method as early as 1795, when he was 18 years old. Again, he adopted it in 1801 to calculate the orbit of the newly discovered planet Ceres. Gauss published his way of looking at the least-squares approach in 1809 and gave several hints that the least-squares algorithm was a minimum variance linear estimator and that it was derivable from maximum likelihood considerations. Laplace wrote a very substantial chapter about the method in his fundamental treatise on probability theory published in 1812.
文摘The purpose of this paper is to introduce second order (K, F)-pseudoconvex and second order strongly (K, F)- pseudoconvex functions which are a generalization of cone-pseudoconvex and strongly cone-pseudoconvex functions. A pair of second order symmetric dual multiobjective nonlinear programs is formulated by using the considered functions. Furthermore, the weak, strong and converse duality theorems for this pair are established. Finally, a self duality theorem is given.
文摘As known, the method to obtain a sequence space by using convergence field of an infinite matrix is an old method in the theory of sequence spaces. However, the study of convergence field of an infinite matrix in the space of almost convergent sequences is so new （see [15]）. The purpose of this paper is to introduce the new spaces ^ ～f and fo consisting of all sequences whose Ceshro transforms of order one are in the spaces f and ^ ～ f0, respectively. Also, in this paper, we show that ^ ～f and ^ ～f0 are linearly isomorphic to the spaces f and f0, respectively. The β- and γ-duals of the spaces ^ ～f and 2% are computed. Furthermore, the classes （^ ～f： μ） and （μ ： f） of infinite matrices are characterized for any given sequence space μ, and determined the necessary and sufficient conditions on a matrix A to satisfy Bc-core（Ax） K-core（x）, K-core（Ax） Bg-core（x）, Bc-core（Ax） Be-core（x）, Bc-core（Ax） t-core（x） for all x ∈ t∞.
文摘In this paper, the extremal problem, min, of two convex bodies K and L in ?n is considered. For K to be in extremal position in terms of a decomposition of the identity, give necessary conditions together with the optimization theorem of John. Besides, we also consider the weaker optimization problem: min. As an application, we give the geometric distance between the unit ball B2n and a centrally symmetric convex body K.
文摘Several general open mapping theorems for continuous linear maps, weakly continuous linear maps and closed linear maps are established respectively. From these general forms, a lot of specific open mapping theorems can be deduced.
文摘In this paper, the author establishs general open mapping theorems for continuous, weakly continuous, closed and weakly singular linear maps respectively. From these general theorems, he deduces a lot of specific open mapping theorems, which include some well-known theorems and some new interesting results. Particularly the author gives the extensions of Husain's open mapping theorem and Adasch's open mapping theorem.
