相关论文: Multi-dimensional vector product
Vector products can be defined on spaces of dimensions 0, 1, 3 and 7 only, and their isomorphism types are determined entirely by their adherent symmetric bilinear forms. We present a short and elementary proof for this classical result.
The definition of vector cross product (VCP) introduced by Eckmann only exists in thethree- and the seven- dimensional vector space. In this paper, according to the orthogonal completeness, magnitude of basis vector cross product and all…
It is shown that scalar product of two vectors can be introduced in any geometry (metric space) independently of possibility of the linear space introduction. In general, linear properties of scalar product are restricted. Domain of…
The notion of virtual global generation (VGG) for a vector bundle has multiple possible generalization from the case of curves to higher dimensional normal projective varieties. We study relationship between these notions. All these notions…
Although the vectorization operation is known and well-defined, it is only defined for 2-D matrices, and its inverse isn't as well-popularized. This work proposes to generalize the vectorization to higher dimensions, and define…
We demonstrated using an elementary method that the inertia tensor of a material point and the cross product of two vectors were only possible in a three or seven dimensional space. The representation matrix of the cross product in the…
Two successive generalizations of the usual tensor products are given. One can be constructed for arbitrary binary operations, and not only for semigroups, groups or vector spaces. The second one, still more general, is constructed for…
Vector calculus in three-dimensional space is ubiquitous in applications of mathematics in physics and engineering. Its two-dimensional version is, however, quite rare. Here we try to provide a pedagogical account of the subject. It is…
A surprising simple result about quadrilaterals is given as an application of the vector triple product identity.
In this paper, we generalize an elementary real-analysis result to a class of topological vector spaces. We also give an example of a topological vector space to which the result cannot be generalized.
We prove that every automorphism of an infinite-dimensional vector space over a field is the product of four involutions, a result that is optimal in the general case. We also characterize the automorphisms that are the product of three…
It is probably safe to say that just about everyone reading this article is familiar with the cross product and the dot product. However, what many readers may not be aware of is that the familiar properties of the cross product in three…
An analytic proof is given which shows that it is impossible to extend any triple of mutually unbiased (MU) product bases in dimension six by a single MU vector. Furthermore, the 16 states obtained by removing two orthogonal states from any…
In the present paper, a notion of M-basis and multi dimension of a multi vector space is introduced and some of its properties are studied.
We discuss the question when a finite-dimensional diffeological vector space is, or turns out not to be, the coproduct of its subspaces in the category of diffeological vector spaces, after reviewing the same question in some other…
In textbooks and historical literature, the cross product has been defined only in 2-dimensional and 3-dimensional Euclidean spaces and the cross product of only two vectors has been defined only in the high dimensional Euclidean space…
Let $D$ and $E$ be subspaces of the tensor product of the finite-dimensional Hilbert spaces $\mathbb{C}^m \otimes \mathbb{C}^n$. We show that the number of product vectors in $D$ with their partial conjugates in $E$ is uniformly bounded…
It is shown that certain transformations on quiver-dimension vector pairs induce isomorphisms on the corresponding moduli spaces of quiver representations and map a stable dimension vector to a stable dimension vector. This result combined…
We show that a complete set of seven mutually unbiased bases in dimension six, if it exists, cannot contain more than one product basis.
In three-dimensional Euclidean geometry, the scalar product produces a number associated to two vectors, while the vector product computes a vector perpendicular to them. These are key tools of physics, chemistry and engineering and…