Related papers: Extendable orthogonal sets of integral vectors
Multiple orthogonal polynomials are a generalization of orthogonal polynomials in which the orthogonality is distributed among a number of orthogonality weights. They appear in random matrix theory in the form of special determinantal point…
Dual quaternions have gained significant attention due to their wide applications in areas such as multi-agent formation control, 3D motion modeling, and robotics. A fundamental aspect in dual quaternion research involves the projection…
The purpose of this effort is to investigate if the use of quaternion mathematics can be used to better model and simulate the electromagnetic fields that occur from moving electromagnetic charges. One observed deficiency with the commonly…
We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra. As applications, we consider generalized inverses, eigenvalues and eigenvectors, similarity, determinants of complex…
This is a review of ($q$-)hypergeometric orthogonal polynomials and their relation to representation theory of quantum groups, to matrix models, to integrable theory, and to knot theory. We discuss both continuous and discrete orthogonal…
We study orthogonal and symplectic matrix models with polynomial potentials and multi interval supports of the equilibrium measure. For these models we find the bounds (similar to the case of hermitian matrix models) for the rate of…
Some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored.
We present a useful proposition for discovering extended Laplace-Runge-Lentz vectors of certain quantum mechanical systems. We propose a new family of superintegrable systems and construct their integrals of motion. We solve these systems…
In this study, we introduce the concept of commutative quaternions and commutative quaternion matrices. Firstly, we give some properties of commutative quaternions and their Hamilton matrices. After that we investigate commutative…
We study directions along which the norms of vectors are preserved under a linear map. In particular, we find families of matrices for which these directions are determined by integer vectors. We consider the two-dimensional case in detail,…
The Eigendecomposition of quadratic forms (symmetric matrices) guaranteed by the spectral theorem is a foundational result in applied mathematics. Motivated by a shared structure found in inferential problems of recent interest---namely…
This paper presents an experimental study on the application of quaternions in several machine learning algorithms. Quaternion is a mathematical representation of rotation in three-dimensional space, which can be used to represent complex…
We study experimentally systems of orthogonal polynomials with respect to self-similar measures. When the support of the measure is a Cantor set, we observe some interesting properties of the polynomials, both on the Cantor set and in the…
As an expansion of complex numbers, the quaternions show close relations to numerous physically fundamental concepts. In spite of that, the didactic potential provided by quaternion interrelationships in formulating physical laws are hardly…
The field of deep learning has seen significant advancement in recent years. However, much of the existing work has been focused on real-valued numbers. Recent work has shown that a deep learning system using the complex numbers can be…
A new approach to normal operators in real Hilbert spaces is discussed, and a spectral representation is obtained, derived directly from the complex case. The results are then applied to quaternionic normal operators, regarded as a special…
We carry further our work [DV2] on orthonormal integrators based on Householder and Givens transformations. We propose new algorithms and pay particular attention to appropriate implementation of these techniques. We also present a suite of…
This paper revisits the notion of classical orthogonal polynomials from a broader functional-analytic point of view. It is intended neither as a survey of known results nor as a review of the literature, but rather as a conceptual…
We study the orthogonal polynomials associated with the equilibrium measure, in logarithmic potential theory, living on the attractor of an Iterated Function System. We construct sequences of discrete measures, that converge weakly to the…
An involution is usually defined as a mapping that is its own inverse. In this paper, we study quaternion involutions that have the additional properties of distribution over addition and multiplication. We review formal axioms for such…