Related papers: Division and new multiplication between vectors
In engineering practice one often encounters planar problems, where the corresponding vector space of forces, velocities or (infinitesimal) displacements is three dimensional. This paper shows how these spaces can be factorized, such that…
Typically, when we are given the section (or projection) function of a convex body, it means that in each direction we know the size of the central section (or projection) perpendicular to this direction. Suppose now that we can only get…
We introduce two generalizations of bracket vectors from binary trees to permutrees. These new vectors help describe algebraic and geometric properties of the rotation lattice of permutrees defined by Pilaud and Pons. The first…
This paper revisits the little-known Gibbs-Rodrigues representation of rotations in a three-dimensional space and demonstrates a set of algorithms for handling it. In this representation the rotation is itself represented as a…
Different (not only by sign) affine connections are introduced for contravariant and covariant tensor fields over a differentiable manifold by means of a non-canonical contraction operator, defining the notion dual bases and commuting with…
Image tracing is a foundational component of the workflow in graphic design, engineering, and computer animation, linking hand-drawn concept images to collections of smooth curves needed for geometry processing and editing. Even for clean…
We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the Lambda-calculus…
This article demonstrates that convolutional operation can be converted to matrix multiplication, which has the same calculation way with fully connected layer. The article is helpful for the beginners of the neural network to understand…
Diagrams as a graphic expresion of derivatives is proposed for calculation of derivatives for composed function. The concret diagram is understood as a virtual derivative in contrast of concret derivative. In polynomial expression of…
In this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and…
In a previous paper we have introduced a class of multiplications of distributions in one dimension. Here we furnish different generalizations of the original definition and we discuss some applications of these procedures to the…
A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a…
We provide a characterization of the finite dimensionality of vector spaces in terms of the right-sided invertibility of linear operators on them.
Distributional semantics creates vector-space representations that capture many forms of semantic similarity, but their relation to semantic entailment has been less clear. We propose a vector-space model which provides a formal foundation…
A complete classification is obtained of continuous, translation invariant, Minkowski valuations on an m-dimensional complex vector space which are covariant under the complex special linear group.
We characterize bijections on matrix spaces (operator algebras) preserving full rank (invertibility) of differences of matrix (operator) pairs in both directions.
In this note, we provide a important considerations of a familiar topic: the gradient of a vector field. The gradient of a vector field is a common quantity represented in continuum mechanics. However, even for Cartesian coordinate systems,…
We introduce the notion of one-sided mapping cones of positive linear maps between matrix algebras. These are convex cones of maps that are invariant under compositions by completely positive maps from either the left or right side. The…
An effective method of computing division polynomials in terms of Mumford coordinates is presented. As an example, division polynomials for $3$- and $4$-torsion divisors on a genus two curve are obtained explicitly in terms of Mumford…
Vertex splitting is a graph modification operation in which a vertex is replaced by multiple vertices such that the union of their neighborhoods equals the neighborhood of the original vertex. We introduce and study vertex splitting as a…