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[4] viXra:2311.0146 [pdf] submitted on 2023-11-28 05:53:23
Authors: Chengshen Xu
Comments: 8 Pages.
In this paper, we prove that the sufficient conditions for the extreme value problem with constraints requires that the projection of the gradient of the point in the final constraint surface is zero and that the second-order partial derivative matrix--Hessian matrix on the local linear subspace in the constraint surface is positive definite or negative definite. The necessary conditions require that the projection of the gradient of the point in the final constraint surface be zero and that the second-order partial derivative matrix--Hessian matrix on the local linear subspace in the constraint surface be semi-positive definite or semi-negative definite. Finally, we discuss the reconstruction of the micro-base vector on the local linear subspace in the constraint surface and the local coordinate system.
Category: Algebra
[3] viXra:2311.0145 [pdf] submitted on 2023-11-28 06:00:52
Authors: Chengshen Xu
Comments: 8 Pages.
A real matrix may not be similar to a diagonal matrix or a Jordan canonical matrix in the real number field. However, it is valuable to discuss the quasi-diagonalization and quasi-Jordanization of matrices in the field of real numbers. Because the characteristic polynomial of a real matrix is a real coefficient polynomial, the complex eigenvalues and eigenvector chains occur in complex conjugate pairs. So we can re-select the base vectors to quasi-diagonalize it or quasi-Jordanize it into blocks whose dimensions are no larger than 2. In this paper, we prove these conclusions and give the method of finding transition matrix from the Jordan canonical form matrix to the quasi-diagonalized matrix.
Category: Algebra
[2] viXra:2311.0140 [pdf] submitted on 2023-11-28 21:53:25
Authors: Ebubekir Kaya
Comments: 6 Pages. In English and German
The suggestion real and imaginary rectangles is an extension of complex numbers. With this representation, polynomial functions can be visualized. And there is a special relationship between these functions.
Category: Algebra
[1] viXra:2311.0112 [pdf] submitted on 2023-11-24 02:13:15
Authors: Juan Elias Millas Vera
Comments: 4 Pages.
In this paper I will apply some of notation tools in hyperoperators to the understanding of a deep and developed theory in notation. Using [a] or [-a] for an a Natural we will see the richness of this algebraic writing form.
Category: Algebra
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