Related papers: On the generalized vectorization and its inverse
Given an automorphism and an anti-automorphism of a semigroup of a Geometric Algebra, then for each element of the semigroup a (generalized) projection operator exists that is defined on the entire Geometric Algebra. A single fundamental…
The purpose of this article is to study certain binary relations of endomorphisms over infinite dimensional vector spaces defined by GD1 and 1GD generalized inverses. In order to do so, these generalized inverses are studied over arbitrary…
Inverse optimization describes a process that is the "reverse" of traditional mathematical optimization. Unlike traditional optimization, which seeks to compute optimal decisions given an objective and constraints, inverse optimization…
Projection matrices are necessary for a large portion of rendering computer graphics. There are primarily two different types of projection matrices -- perspective and orthographic -- which are used frequently, and are traditionally treated…
Convenient parameterizations of matrices in terms of vectors transform (certain classes of) matrix equations into covariant (hence rotation-invariant) vector equations. Certain recently introduced such parameterizations are tersely…
The paper explains how a unit generalized quaternion is used to represent a rotation of a vector in 3-dimensional space. We review of some algebraic properties of generalized quaternions and operations between them and then show their…
Vector representations of graphs and relational structures, whether hand-crafted feature vectors or learned representations, enable us to apply standard data analysis and machine learning techniques to the structures. A wide range of…
It is well-understood that the robustness of mechanical and robotic control systems depends critically on minimizing sensitivity to arbitrary application-specific details whenever possible. For example, if a system is defined and performs…
This article studies the inverse problem of recovering a vector field supported in $\mathbb{D}_R$, the disk of radius $R$ centered at the origin, through a set of generalized broken ray/V-line transforms, namely longitudinal and transverse…
This study focuses on convex functions and their generalized. Thus, we start this study by giving the definition of convex functions and some of their properties and discussing a simple geometric property. Then we generalize E-convex…
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…
It is sometimes desirable to produce for a nonlinear system of ODEs a new representation of simpler structural form, but it is well known that this goal may imply an increase in the dimension of the system. This is what happens if in this…
Invariant coordinate selection is an unsupervised multivariate data transformation useful in many contexts such as outlier detection or clustering. It is based on the simultaneous diagonalization of two affine equivariant and positive…
In this paper will be considered standard forms of generalized inverses for matrices in the shape of block representations {1, 2, 3, 4, 5, 5^k}-inverse. Especially will be considered Moore-Penrose inverse and the group inverse. Results from…
We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: Regularization Graphs. Regularization graphs allow to construct functionals using as building blocks linear…
Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from…
We present two generalisations of Singular Value Decomposition from real-numbered matrices to dual-numbered matrices. We prove that every dual-numbered matrix has both types of SVD. Both of our generalisations are motivated by applications,…
We introduce a rotation-invariant representation of planar shapes. In particular, this representation encodes shapes as vectors such that the Euclidean distance between them serves as a valid shape distance. For standardized, star-shaped…
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…
It is interesting to know, how far we can generalize the notion of a group-valued cocycle keeping the property to determine a bundle. We find a generalization for pairs of cocycles and show how these generalized pairs of cocycles can still…